There were a few purposes to strategic bombing. The first was to destroy the enemy's ability to manufacture war materials, usually by destroying a key piece in the chain of production. For instance, the American raids on Schweinfurt targeted ball bearing production and aircraft production. If the enemy was unable to make ball bearings then almost nothing that was engine-powered would work. The enemy's capacity to make war would literally grind to a halt.
The second objective was often to kill and terrify the enemy populace. Sometimes civilian casualties happened despite the best efforts of bombers to hit factories or infrastructure, but often civilians were specifically targeted as well. The concept was that once the civilians saw what could happen to them then support for the war would cease, and perhaps the civilians would demand that the war be brought to a stop.
Finally, one could use strategic bombing to force the enemy air force to make a difficult choice: either allow the enemy to bomb your territory at will or come up to challenge your planes in the air and risk aerial combat. This is what happened to the Luftwaffe, which was destroyed largely in the air once the Americans were able to deploy large numbers of long-range escort fighters for their bombers in 1944.
The first idea of hitting key enemy factories only had moderate success. Bombers were wildly inaccurate, especially when they were bombing from high altitude (which was done to avoid as much anti-aircraft fire as possible). Many bombs hit nowhere near their intended targets. Other times the bombers had navigation difficulties (often at night) and didn't even bomb the right city, much less a particular factory in that city. Further, industry proved to be more resilient than bomber advocates hoped. One could disperse one's factories so that if one was lost then the others could continue production, factories could be rebuilt, and you could build up stockpiles of product so that production could continue despite a pause in production of one particular part. The second idea of killing and terrifying the enemy populous turned out to be problematic as well. Civilian morale proved to be as resilient as enemy industry, and often the determination of those being bombed would grow rather than wither. The last idea of forcing the enemy's air force to defend their territory was a disaster for the Germans in the Battle of Britain, but was an eventual success for the Americans in 1944--leading to a Luftwaffe that could not effectively oppose the D-Day landings or the advance of the British and American armies in the west.
Strategic bombers were used to hit targets that were not deemed to be strategic. The most famous of these was the Transportation Plan, in which the transportation infrastructure around northern France was repeatedly hit prior to D-Day. These attacks were costly, and they were supplemented with fighter-bomber attacks and strikes from medium bombers. It took repeated attacks to accomplish the goal of disrupting German movement, and it often took follow-on attacks once the Germans repaired the rail lines and bridges that were hit.
Also, there was the tactical use of bombers such as in Operation Cobra that did allow for the Allied breakout from Normandy. Front line German positions were hit by Allied bombers, and the devastation was immense. However, there was also a tragic amount of friendly fire (including the highest-ranking US soldier to be killed in Europe, Lieutenant General Lesley McNair). These attacks were possible because the front line was relatively static--allowing such a large strike to be planned and executed. When bombers had a hard time hitting a particular city sometimes, hitting the right target when friendlies are very close was even more difficult. Cobra was a success, but the specter of friendly fire loomed large.
I hope that answers your question. As always, followup questions from OP and others are encouraged.
I have a follow up regarding the third "goal" you list; I reread a book recently that we used in high school (Why The Allies Won, idk the author as I'm at work) and when talking about the strategic bombing campaigns one of the benefits he cites is that once strategic bobbing began in earnest, such a large portion of the Luftwaffe fighter force was brought back to Defend Germany that the Germans lost their air supremacy on the eastern front. Would you say this is true, and how much impact did it have?
Actually going off of this another point made in that book is that a great deal of war production of some items (artillery, optics, radio and communications gear) was also devoted to air defense of Germany, if you know anything about that and would agree or disagree I'd be interested to hear your (or anyone else's) thoughts