There is a lot to unpack here, much of it having to do with geography, more immediate military and political goals of warfighting, the realities of planning to fight a global war, and that in the end the other side gets a vote too.
We should also note that there is some basic inaccuracy in your question OP, or at least it is an imbalanced statement.
The first is that due to internal political calculations on both sides, Germany was actually first to declare war on the US on December 11th, followed then quickly by a reciprocal from the US.
And as by any definition there were American servicemen and German servicemen firing at each other well before the start of 1943, indeed even before the official declaration of war. On 3 occasions in the Fall of 1941 US Navy destroyers escorting convoys as part of the Neutrality Patrol engaged or were engaged by German subs. On September 4th it was the USS GREER, October 17th it was the USS KEARNY, and on October 31st it was the USS REUBEN JAMES that was the first USN warship lost in combat in WW2. From the first attack on the USN ships assigned to convoy duty were not just working to help make up for the lack of British ships on the first half of a convoy's journey, but were actively interposing themselves and guarding, while operating under orders to shoot on site. So just as US warships were fighting the war against Japan from the outset so too were they the first contribution of the US in the War in Europe and Africa.
It is also worth again recognizing the realities of US dispositions for fighting the war when it came and how that played into the path 1942 took. While a Europe First strategy was generally accepted by most, that did not then equate passivity in the Pacific, and even setting aside and accepting the loss of those forces forward deployed in places like the Philippines, there were still targets the US and Allies considered too important not to fight for. Among those were the sealanes through the Coral Sea that linked the US to Australia, key to any future buildup and counterattack, and any island further beyond 180th Meridian /International Date Line, which would be a base able to strike at vital sealanes or even Hawaii, among those was then Midway. So while buildup of forces was generally limited, and transfers from the Atlantic to the Pacific were made mostly to makeup losses and to prevent further erosion, that did not mean that Admirals King and Nimitz intended to fight a reactive war.
And while all the USN was fighting a bitter war they were poorly prepared for in the Atlantic against the U-Boats, there wasnt really any way to engage those German surface forces left in mid 1941.
And for all the ballyhoo about Midway as a true turning point(a matter of debate), and one way or the other it did buy the Allies needed operational breathing room and parity, the real instance of taking the fight to the Japanese wouldnt be until August of 1942 and Operation WATCHTOWER, the invasion of Guadalcanal and surrounding islands. And even then the exact timing of the invasion was rushed in response to an increased Japanese presence in building an airfield on the island, with an initial landing force built around the 1st Marine Division and any other unit that could be gathered together, even then the forces to be put ashore were only about 15% the size of the TORCH landing forces to hit North Africa a little over two months later (15k vs 100k). And beyond numbers was in almost all things an invasion run on a budget that was on very thin margins for almost every way for several months.
While on the other side of the globe there had been months of debate about how best to strike with the new allied force in Europe, almost from the first day of the US officially joining the war. Churchill and his military chiefs flew to Washington for the ARCADIA Conference two weeks after Pearl Harbor and over the next month the Allies began to form the official bodies and organizations to coordinate conduct of the war, among them the official combined Joint Chiefs, the ill fated ABDACOM in the Pacific, and the identification of targets and priorities. It was also recognized that none of this could happen immediately, the US military had begun more aggressive expansion in 1940/41 but still needed months to years for new units to be organized, filled with men, equipped, trained, and shipped to where they would be needed in numbers to actually matter. And here there was some debate and differences of opinion, much of the US military leadership wanted to focus on building up in the UK, with the aim of getting men ashore in France as quickly as they could. Various versions of the plan depending on the timing(late 1942 if the USSR was falling, mid 1943 more likely) were called SLEDGEHAMMER and ROUNDUP, with the general buildup of USAAF forces called BOLERO. The British though were less crazy about the plan, as it would involve a large % of British troops that were increasingly drawing upon limited manpower, and with Allies who they considered amateurs at most at that point in the war.
Presenting a very united front the British Chiefs, including Churchill appealing directly with FDR, were able to also point the way to North Africa, where Allied Naval and Air forces could interdict the reinforcement of the joint German Italian forces facing Montgomery and the British 8th Army, while Vichy North Africa could either be taken relatively easily, or more hopefully brought wholesale into the fold. The Mediterranean could be made more secure as an incredibly valuable shipping route, the US forces could gain experience, and British troops wouldnt get killed trying to land in France only to be thrown back. By mid 1942 doing TORCH first had won out and while buildup in the UK continued forces were being earmarked for Africa and plans drawn up. And so after a mass coordinated movement of multiple invasion forces from the East Coast of the US and British Isles and the landings went off on November 8th 1942. But the enemy was only the forces of Vichy France and actual fighting was brief if violent in many places, and the Allied forces pressed on towards Tunisia and the heart of Axis forces in Africa.
There on 28th/29th of November 1942 was the first time the US Army Ground Forces would fight German Army units, when units including Combat Command B of the 1st Armored clashed with elements of the 10th Panzer Division. Attacks through the end of 1942 including what came to be called Longstop Hill were mostly ineffective against the joint Italian-German forces and left the Allies in essentially the same positions they began in. Setting the stage for German counterattacks at places like Sidi Bou Zid and Kasserine in February 1943 before being stopped, and a reorganized and reinforced Allied force began the final encirclement of the Axis forces from both West and South with Tunis finally falling in early May.
In the end the first ground combat between US forces and those of Japan and Germany in the first true offensives launched against each were only separated by about 3 months(mid August vs late November). And is easily explained by the need to decide on a target, gather sufficient forces for the task, and actually get to where the enemy was, and the rushed near ramshackle nature of the first Allied landings in the Pacific especially.
If you want some good next places to consider reading there are 3 books I would recommend to start your additional reading:
An Army at Dawn by Atkinson
Pacific Crucible and Conquering Tide by Ian W. Toll
All 3 are top notch reads, and solid history without being too dense for someone not stepped in the subject matter and are parts of larger series on the US Army in Europe and the War against Japan respectively.
u/DBHT14 has already noted that the United States was quite involved in the struggle against Germany prior to entering the war against German in December 1941. Here are a few highlights of American involvement in the first half of WWII.
Now, getting to your question. You asked: "why did it take so long to engage the Germans in the war?" As we can see, the Americans were engaging the Germans well before the war started!
But by "engage the Germans" I think you're asking this question: Why did it take so long for the Allies to invade Italy and France? There are many reasons for this.
Factor 1: The U.S. armed forces needed time to build up.
In 1940, the U.S. Army had only about 180,000 men (if you account for the National Guard and reserves, then it's about 270,000 troop). That made it only the 19th-largest army in the world, smaller than Romania and Portugal! American soldiers and airmen were still equipped with obsolete and obsolescent weapons and equipment.
The Selective Training and Service Act of 1940, which was signed into law in September 1940 helped swell the Army's ranks with draftees, giving the Army 1.45 million men by the outbreak of war in 1941. However, many of these men were still undergoing training. Well into 1942 and 1943, the U.S. Army was still forming and training up infantry and armored divisions. It took a lot of time to train men and build a division. Soldiers didn't just have to learn their jobs, they had to learn to work as a team. Artillery battalions had to learn how to support infantry battalions. Tank platoons had to learn how to work with infantry platoons. For much of late 1942, the vast expanses of the Mojave desert were criss-crossed by tanks and sweating infantrymen drilling for the invasion of North Africa. All in all, it took *years* to forge enough divisions to invade Italy in mid-1943 Italy and land in Occupied France in mid-1944.
Factor 2: Allied agreements.
At the December 1941-January 1941 ARCADIA Conference in Washington (the same conference that affirms the "Germany first" strategy). Senior American and British leaders reached the following agreement (emphasis added):
"In 1942, the methods of wearing down Germany's resistance will be . . . ever increasing air bombardment by British and American forces . . . assistance to Russia's offensive by all available means . . . [and operations] the main object [of which] will be gaining possession of the whole North African coast ... It does not seem likely that in 1942 any large scale land offensive against Germany, except on the Russian front, will be possible ... [but] in 1943, the way may be clear for a return to the continent across the Mediterranean, from Turkey into the Balkans, or by landings in Western Europe. Such operations will be the prelude to the final assault on Germany itself."
From the very onset, Allied planners realized that a pre-1943 attack would be very, very difficult to pull off. Britain was weak, Germany was strong and American had yet to marshal all its strength.