I was thinking about starting out as just the soldier class. Should I or are the other classes more fun to play?
The problem with Internet quotations is that many are not genuine. - Abraham Lincoln
The most fun class for me is the Vanguard as ramming into your enemy headfirst, shooting them in the face with a shotgun, and then elbowing them in the head. Then doing it all over again. Vanguards are one of the most entertaining one-trick ponies in gaming if you ask me.
Still Soldier isn't a bad choice just don't get caught in the trap of using only one gun, you have Assault Rifles, Shotguns, and Sniper Rifles use all 3 of them. Change up your tactics, that makes Soldier infinitely more entertaining than sitting around and plinking away at people with only one weapon.
Still if I were to rate the classes in terms of most fun to least fun (obviously this is my opinion)
"I love nailing Asari, so ageless and superior. Then you get them and they squeal like schoolgirls"- Gianna Parasini, Mass Effect 2.
Every class has elements that can make them fun.
Soldiers have a versatile weapon loadout so they can go from short-range to medium-range to long-range combat on a whim. With Adrenaline Rush and a powerful enough weapon you can take down multiple powerful enemies in a short span. However I like using powers periodically and primarily being stuck with weapons can get a little dull after a while.
As Aigonroth said Vanguard can be a very entertaining class. Charging an enemy and blasting in the face with a shotgun is both satisfying and exhilarating. But it is also dangerous too. Charge at the wrong time and you can find yourself quickly surrounded by enemies. I died more times playing a Vanguard than any other class.
Infiltrators are also very fun. For the first few seconds an Infiltrator looks through a sniper rifle scope time is slowed down. It is a very satisfying feeling to carefully line up headshots on enemies and then blow them away. The Tactical Cloak ability is a fun and useful ability that lets you set up a sniper perch or escape enemies if mobbed.
Sentinels are walking tanks. Their Tech Armor ability greatly strengthens your shields and can be used to instantly replenish them if they go down. And with access to Warp and Overload they can singlehandedly strip away any enemy defense.
Adepts have such a wide variety of biotic powers that you can forget you even have a weapon. Their real strength is in biotic combos. Trap an enemy Singularity or Pull and then hit them with Warp to create an explosion, killing the trapped enemy and damaging any others in the area.
Engineers have Overload and Incinerate to weaken enemies with and Combat Drone is an effective crowd control ability. However I think this is the class I least enjoy playing as. It has useful abilities and can be powerful but just doesn't feel as exciting as the rest.