Well first of all, welcome to the Bowser Boards. I always love seeing new faces around here, especially those who love Bowser the way we do. I am going to try and write a short, comprehensive guide for you and explain his movement and moves.
Moving With Bowser
This is one of the primary aspects of Bowser we have been analyzing together lately. Having consistent, fluid movement with Bowser is possibly more important than with any other character. Without it, you are a sitting rock, and you're gonna get bodied hard. Learning to wavedash and waveland consistently is going to be one of the first things you'll want to make sure you can do. After that, Fortress Hogging is important for grabbing ledge in many situations, but I will go over more things later. Wavelanding on platforms gives Bowser a hell of a lot of mobility, so make sure that you can do that, as it is useful for both escaping, approaching, and generally applying unpredictable, slippery movement.
There are a few tactics that we have been experimenting with that will take a lot longer to learn. One of these is Bowser's moonwalk. Bowser can only moonwalk out of a wavedash or waveland, and the timing can be very difficult to learn. Once you do, however, you'll be able to move around much faster from platform to platform, and you can use it to grab the ledge from farther away than usual while already facing the ledge so you don't have to Fortress Hog.
Bowser Basics
So, now you can move. Cool, what's next?
I am going to go over the moves that you will be likely using the most, and then briefly talking about the others, and why they suck big nuts.
F-Air
This will likely be the most you use the most. The range on it is very good, and the knockback is good as well. This is going to be one of your primary spacing tools, as well as one of your better combo moves. This is one of his best edgeguarding moves as well, having a fantastic knockback angle and hitbox that can reach slightly below the ledge if you time if correctly with a short hop.
Up-B
You probably already know about this. Bowser's upB OoS is one of the best out of shield options in the game, coming out instantly and being incredibly strong. At high percents it can even be used as a kill move, especially on floatier characters. In my opinion, you do not want to overuse this move, as people will begin to just avoid it, and it can be highly punishable on miss or if the weak hitbox connects.
Side-B (Koopa Klaw)
This is the move that separates Bowser from the rest of the cast, his command grab. It is actually very difficult to use against players that are used to it, so you have to create setups for it. I will go over a few of those later. KKfthrow is one of his best combo starters on fast fallers (though it isn't nearly as useful against floaties), and KKbthrow is a fantastic kill move and is great for edgeguarding characters like Marth when they recover above the ledge.
U-Air
This is my favorite move in the game, and I think it is one of Bowser's best tools. You can apply a sort of mental pressure with this, since it is such a scary move. You can use it to tech chase underneath platforms, which can combo into move uairs, a fair, bair, or even a KK if you can pull it off, though it is difficult to do so. Uair is incredibly powerful, killing most characters early, and killed floaties very early. Using this as an anti-air against enemies who are coming down is a great way to net kills early.
F-Tilt
Bowser's ftilt is fantastic for edgeguarding. The hitbox is much larger than most people expect, and the knockback is insane at high percents. It isn't a bad move for spacing either, but don't rely on it because it is still punishable. It's a great move for covering rolls, or making reads. When you combine ftilt with a perfect wavedash, it's scary for the enemy because you can cover so much ground with it.
U-tilt
This is one of Bowser's best kill moves, primarily on floaties. It is actually better than upsmash, both in frame data and knockback. It comes out extremely fast, and has a lot of knockback. It isn't quite as useful at low percents on fast fallers, though, so keep in mind that it is possible to be punished off of it early on. Something important to remember is that jab > utilt is a true combo on floaties, and this is one of your best kill confirms.
N-Air
Bowser's nair is good, though I don't have particularly much to say about it, myself. The hitbox is good as it covers your body, and it comes out relatively quickly. This move is good for edgeguarding from ledge (invincible nair) and is also useful when you are coming down as a "get off me" move.
B-Air
This move is almost exclusively going to be used for edgeguarding, as it is incredibly laggy and punishable. It can be crouch cancelled on stage, or simply shielded. However, when it comes to edgeguards, this move is very useful. Invincible bair from ledge is a great way to send them back off with no chance of recovery. You can also use a double jump bair to catch people off guard who are trying to recover high.
F-Smash
This move is somewhat of an anomaly. Regarding frame data, this move is terrible. However, the knockback is so strong that it is in your best interest to pull it out occasionally for a read or an edgeguard. It is actually a very useful edgeguarding tool on some characters, such as spacies and Falcon. Keep this move in your pocket, and save it for a special occasion, as it is extremely easy to punish when it whiffs.
D-Smash
This is a very strong multihit that is useful in multiple situations. You can crouch cancel into it (occasionally, do not rely on this working every time), read a roll in, or use it to edgeguard. This can be used as a kill move as well.
Now for the moves that aren't so useful. I'll keep these short.
D-Air
This is a poor multihit move that doesn't do much. Somewhat useful for coming down if you can ledge cancel reliably, but other than that, not much use.
D-Tilt
Very slow, not very strong. Down angled ftilt is better in every way, use that instead.
Down-B
Bowser Bomb is decent for grabbing ledge sometimes, but that is pretty much it. The hit itself isn't bad, but it is easy to avoid and the ending lag is abysmal.
U-Smash
It's decent sometimes to cover yourself when someone is coming down on you, but in most situations you will be better off using uptilt.
Well, I think that just about covers it for the basics on his moves. You're more than welcome to join the Bowser Discord server I have set up, which Dralro posted a link to above, where several Bowsers discuss various things we find, as well as just hang out. There's Synnett, Dralro, Stockholm Syndrome, A Stuffy Muffin, myself, and a few others. Any other questions you have, just let me know and I'll answer them the best I can.