Ryu is a fairly easy character to get to grips with.
Everyone knows what his special moves do and how to perform them, which is a good start when it comes to learning his moves.
However, knowing what his moves do and how to use them are two different things. Ryu's style leans towards a fast rushdown pace that doesn't give your opponent much room to breathe, so you need to be confident with what you're doing and
Hadouken (quarter-circle towards + P)
Ryu's fireball is good for helping control the screen and slowing the pace of the match down but don't aimlessly throw them out in a predictable pattern. If your opponent jumps over one, you'll be in trouble. Try and use them from mid-range where they're harder to react to and start using EX Fireballs if your opponent is absorbing your fireballs with Focus Attacks, as he will be doing this to build meter.
Be aware that some characters have moves that pass through fireballs (Abel when he has Ultra, Blanka's EX Rolling Attack, Zangief's EX Banishing Flat, etc), so don't use fireballs when characters have these options available to them.
Shoryuken (towards, down, down-towards + P)
Use this move to hit opponents out of the air. Simple enough - if you're fighting an opponent who insists on jumping around everywhere, just stay on the ground and get ready to crank the motion out as soon as they leave the ground. You can tempt opponents into jumping with regular fireballs and by using crouching MK a lot, as it's a tough move to get past on the ground.
Try not to get into the habit of using it when you get up off the ground, as good opponents will expect and block it. If they're using Focus Attacks to absorb the Shoryuken's single hit, use EX Shoryuken to smash through it.
Tatsumaki Senpukyaku (quarter-circle back + K)
Or as everyone calls it, the hurricane kick. You can spot new players because they will aimlessly throw this move out but it's not even remotely close to being safe. You can be hit as you pass over a crouching opponent and after you pass them too.
Instead, using it for certain situations. It's a good (if risky) way of getting out of the corner and also passing through some projectiles, such as Guile's Sonic Boom and Dhalsim's Yoga Fire. The EX Hurricane Kick has Ryu rooted to the spot and is good for finishing combos or chip damage.
Shinku Hadouken (quarter-circle towards, quarter-circle towards + P)
Ryu's standard fireball super. Fits easily on the end of combos and can be super-cancelled from a normal fireball for added style points. Just do two quick fireball motions, so you do quarter-circle towards + P, quarter-circle towards + P - this will unleash a normal fireball immediately followed by the super fireball.
Metsu Hadouken (quarter-circle towards, quarter-circle towards + PPP)
Ryu's Ultra. Harder to combo as it has a slightly longer start-up time - you can tag it on the end of a Shoryuken that has been Focus Attack Dash Cancelled or an EX Fireball in the corner. Never randomly throw it out. Opponents can almost always jump over it due to its deceptively long start-up time and it takes ages before Ryu can recover in time to block.
Basic Combo: Jump in HK, crouching MK xx HP Hadouken
Advanced Combo: HP Shoryuken xx Focus Attack > Dash Cancel > Metsu Hadouken