The problem with trying to find alternative methods to carry out Operation Overlord is twofold. Secondly, the D-Day landings did not go that badly, at least in terms of the landing parts. Losses were lower than expected, and were low on four of the five beaches. Total casualties on Utah Beach may have been ~200, while on Juno, Gold and Sword, they were in the region of 500. It was only on Omaha where casualties reached over a thousand; this was more attributable to good defensive terrain, bad weather and Allied intelligence missing the presence of extra German troops rather than to any failure of tactics. It should also be noted that these numbers include wounded and missing men, and count casualties suffered inland or from artillery fire after the beach had been secured. Secondly, amphibious assaults are really hard to plan. Sometimes, you do just have to throw men at a beach, because all the alternatives are worse. In this answer, I'll run through a number of possible alternatives and explain why they were worse.

The easiest way to minimise casualties during an amphibious assault is simple; land where the enemy isn't. If there's nobody there to shoot at you, then casualties would be minimal. However, this was not practical for the Allies in 1944. The Germans had defensive positions, to a greater or lesser extent along the full length of the coastline of Northern France, Belgium and Holland. Going further afield, to Denmark or into the Bay of Biscay, might have allowed the Allies to evade these defensive positions. However, they were not practical for other reasons. This would greatly increase the range from the UK, reducing the availability of air support and increasing the requirements for merchant shipping. These areas are also tricky from a naval perspective, as they would be hard to clear of German submarines; both areas include German submarine bases. You gain little if you've had next to no casualties in the landing, but lost thousands when their troopships were sunk by a submarine. These factors meant that any amphibious assault on France (outside the Mediterranean) would have to come close to the UK, on the Channel coast. The obvious landing ground, the Pas de Calais, which was closest to the UK, had very heavy German defences. The coast between Calais and and Le Havre had very poor terrain, with cliffs overlooking many of the beaches. Going further west into Brittany would stretch the capabilities of Allied fighters, especially Spitfires. Brittany also had a big problem; as a peninsula, the Germans could easily bottle up the landed force inside of it. Normandy, meanwhile, was close to the UK, had largely unfinished defences, and largely favourable terrain. It was the best choice by far.

So, we have to land in Normandy, how about trying to destroy the defences before landing? The Allies did carry out a naval and air bombardment before the landings on June 6th 1944, but this was intended more to keep the defenders' heads down. The aerial bombings were largely inaccurate, and did little. The naval bombardment did succeed in destroying a few bunkers and artillery positions, but again, their main contribution was to suppress the defenders due to their inaccuracy. One way of getting around the accuracy issue would be to extend the length of the bombardment. However, this very obviously telegraphs where the landing is going to take place - the Germans just have to look where the week or month-long barrage has been going on, and move reinforcements there to neutralise the landings. The US was able to carry out long pre-invasion bombardments in the Pacific, against islands that the Japanese could not easily defend. Even with these bombardments, the Japanese were still able to put up heavy resistance, as were the Germans on the Somme in 1916 after a similarly long bombardment. A well dug-in concrete fortification can survive a lot, especially against low-accuracy weapons like naval guns that also fire on a low trajectory. Many of the German defences along the coast were set up with thick concrete or earth walls facing the sea, with embrasures positioned to fire along the beach rather than out to sea. Destroying one of these from the sea would have been impossible, while aircraft bombing was not accurate enough to ensure that every bunker was destroyed in a reasonable timescale. Trying to destroy the defences without guided munitions is hugely impractical.

So, we can't destroy the defences, but what about neutralising them, by landing paratroopers to take them from behind or by using special forces? Paratroopers can land behind the beaches. This made them hugely useful to Allied planners. They could seize the flanks of the landings, and prevent German troops from elsewhere in France from interfering. They could capture key bridges that the Allied advance would need to cross. They could neutralise gun batteries that might threaten the beachhead or the shipping offshore. However, they could not do all of these key tasks and take the beach defenses - though elements of the 82nd Airborne did help to capture the exits from Utah Beach. Paratroopers were not well-suited for landings around beaches; if the pilots' navigation was poor, then the paratroopers could easily end up in the sea. This was not easily survivable for men with parachutes and large amounts of gear. This meant that dropping them further inland made a lot more sense.

Special forces, meanwhile, were not practical as there weren't enough of them. The Allies needed to land on a very broad front, over 75 km of coastline. This made it harder for the Germans to contain the beachhead, and gave the maximum amount of space for the Allies to deploy troops into, allowing them to take advantage of their numbers advantage. However, the concept of special forces was in its infancy. The Allies had almost nothing that might match modern concepts of special forces in the sense of an elite 'door-kicking' raiding force like Seal Team 6 or Delta Force. The closest equivalent would be the SAS, but this was a small force, not able to make an effect over such a broad front. They did have elite light infantry units in the US Rangers and British Commandos. However, these units had their own tasks - the 2nd and 5th Ranger Regiments attacking Pointe du Hoc, the 1st Special Service Brigade landing on Sword to reinforce the 1st Airborne inland, and the 4th Special Service Brigade landing on Sword, Juno and Gold to link up between the three beaches. They were also, as light infantry, poorly suited to attacking defences without support. On the 6th, 48 Royal Marine Commando landed at Langrune sur Mer on Juno Beach, reaching the beach before the infantry and armour that were supposed to have secured it. In taking the beach, they suffered heavy casualties, up to 50%. This was against alerted defences, yes, but they had the benefit of the bombardment.

How about landing at night? Night amphibious landings promise the ability to take the enemy by surprise. However, interwar exercises had shown that this was a false hope. Observers from the shore were able to notice the landing craft surprisingly far from the beach. A night landing also greatly complicates navigation; at the time, this largely relied on observations of landmarks ashore, which were invisible at night. Night landings in the Mediterranean had low casualties, but a lot of confusion due to poor navigation. It also made it much harder for ships and aircraft to bombard the defences. If the assault bogged down, it would be difficult for the troops to receive the fire support they needed to break through the defences.

Another set of ideas that comes up a lot is that of a technological solution. I've addressed the idea of landing craft having doors that open to the rear here and here. /u/DBHT14 has covered the idea of having the infantry push a steel plate across the beach here.

As to what the real plan was for the assault, I will only cover the British (and Canadian) part of it, and leave the American side to another. This started with the naval bombardment, with the largest warships, the battleships and cruisers, engaging the coastal batteries that might otherwise hammer the transports offshore. Bunkers on the beaches themselves were targeted by destroyers, and by gun-armed landing craft. These ranged from the LCG(L), a tank landing craft carrying two naval 4.7in guns, through the LCG(M), a lighter variant of the LCG with two 17pdr AT guns or 25pdr artillery pieces, to the LCS, armed with machine guns and a 6pdr AT gun in a tank turret. There were also a number of rocket firing landing craft, either the LCT(R) firing 5in rockets or the LCA(HR) with the Hedgehog spigot mortar.

Targets on the beachfront itself were engaged by the smaller 'Hunt' class destroyer escorts, while those to the flanks of the beach were hit by larger fleet destroyers. According to the plan, these were to open fire forty minutes before the first troops landed. Ten minutes after the destroyers opened fire, the LCG(L)s and self-propelled artillery in their LCTs were to begin firing on the beachhead. At about the same time, some of the LC(R)s would open fire. As the troops got closer, the LCG(M)s and LCSs would open fire on the beaches. Just before the troops began to disembark, the LCA(HR)s and remaining LCRs would fire on the beaches. However, the other ships and landing craft would begin to cease fire, or shift their fire to targets further in land, to reduce the risk of friendly fire.