NORMAL MOVE COMBOS
Crouching Light Punch, Standing Light Punch, Standing Light Punch, Crouching Heavy Kick ( 147 damage, 210 stun )
This combo is useful if you are not REALLY comfortable with links. The advantage of ending a combo with sweep is that you get an untechable knockdown, giving you ample time to set something up for when they wake up. Also, with Ryu if you hold forward jump after a sweep you get an automatic safejump.
Crouching Medium Punch, Crouching Medium Punch, Crouching Heavy Kick ( 192 damage, 280 stun )
If you are very comfortable with links, this combo is better than the one above because it does more damage and stun. Because Ryu's sweep has such a slow recovery time, using any combo that ends in sweep as a blockstring is not a good idea vs. many opponents because they can punish you before you're recovered. For blockstrings use a combo that ends in fireball.
HURRICANE KICK COMBOS
Jump-in Heavy Kick, Crouching Medium Kick, Medium Hurricane Kick ( 248 damage, 460 stun )
Half the time, you're going to want to replace the cr. med kick with a cr. hard punch, because it will make the combo do more damage and stun. Another big positive for cr. hp is that this move FORCES the opponent to stand up, so even if you start this combo on a crouching opponent, the cr. hp will stand them up, allowing the hurricane kick to hit. The cr. med kick has more range though, so if after the jump in attack you land somewhat far from the opponent then a cr. hp isn't going to connect. Learn the range of when you can use the preferable cr. hp, and when you NEED to use cr. mk. Changing to the cr. hp makes the combo do ( 278 damage, 560 stun )
That said, you can use this combo anytime the first jump in attack is gauranteed to hit (opponent dizzy, jumped over opponents fireball, opponent is in recovery from certain ultras, etc )
Crouching Medium Punch, Crouching Medium Punch, Crouching Medium Kick, Medium Hurricane Kick ( 245 damage, 420 stun )
This combo is pretty hard, like the sweep combo above that similar, you need to be very good at links. It also will not work on a crouching opponent cause the hurricane kick will fly over them. On a crouching opponent you can substitute the hurricane kick for a fireball or EX fireball.
This is high damage and high stun. You can make this a jump-in punish combo, starting it out with a jump-in hard punch (provided you land RIGHT next to the opponent), you can use it after focus crumpling an opponent, or punishing a whiffed move.
Solar Plexus Strike, Crouching Heavy Punch, EX Hurricane Kick ( 338 damage, 460 stun )
This is most viable after you focus crumple an opponent, because the solar plexus is too slow a move for you to throw it out too much randomly, although once in awhile it might catch the opponent off guard as a poke. If done in the corner you can use ultra after the EX hurricane kick.
The solar plexus to cr. hard punch link is super hard, so here's another one where you have to be really good at links.
DRAGON PUNCH COMBOS
Crouching Light Punch, Crouching Light Punch, Heavy Dragon Punch ( 180 damage, 260 stun )
This one is good as a whiff or blocked move punish and also in footsies. Say Hakan slides at you and you block, but you need to hit him with a fast move before he is able to block again, you could do a cr. lp and then dragon punch. This is viable anywhere you can get a cr. lp in whether your move blocks or hits. The second cr. lp gives you time to recognize if the opponent is blocking or being hit, if they are being hit you finish with the dragon punch, if you recognize that they are blocking, finish with a move that is more safe like a fireball.
Solar Plexus Strike, Dragon Punch ( 270 damage, 300 stun )
As for where you can use this, the same rules apply as the combo above that starts with solar plexus, most likely after a focus crumple. This combo, however, is much much easier than the combo above, so you're probably going to want to stick with this one as your default solar plexus combo, unless you're a combo expert.
Crouching Medium Kick, Fireball ( 120 damage, 200 stun )
This is ryu's most important combo/blockstring. Because the cr. mk has good priority, speed, and range, you can walk towards the opponent and throw this out effortlessly in order to keep pressure on them, Daigo-style. It is also almost always safe even if the opponent blocks it. You should be getting some damage/chip damage from this every round.
One thing I like to do is sometimes on an opponents wake up, stay at a range where their dragonpunch/uppercut/reversal move would barely hit you, and block it. This should be close to the edge of the range of your cr. mk, where it's priority is at it's highest. If the opponent does a highly punishable move like a dragon punch, wait for them to land and punish like normal. If they too just sit there blocking, wait just long enough for you to recognize that they're blocking and not doing a reversal move, then throw out a cr. mk into fireball combo. If they try to do a move at the same time as you there's a good chance your cr. mk will beat it or at least trade. If they continue to block, you'll get chip damage from the fireball and not be in any worse a position. A lot of times though, they'll throw out a move a split second after you, or throw out a slow move, and your cr. mk will hit them during the start up frames of their move.
METSU HADOUKEN COMBOS
Dragon Punch, FADC First Hit, Metsu Hadouken ( 398 damage, 150 stun )
Use this any time you anticipate that the opponent is going to do a move that a Dragon Punch will beat, or if you anticipate that they aren't going to be blocking. If they do block, do not finish with your Ultra after the FADC, because they'll just block it and punish you. Instead after the FADC go into a blockstring or do a throw attempt if they blocked the initial dragon punch.
Crouching Medium Kick, EX Fireball, FADC, Metsu Hadouken ( 393 damage, 200 stun )
This is pretty safe to use anytime you have 3 bars of EX meter and Ultra meter charged. If the crouching medium kick is blocked, again don't throw out the ultra, do a blockstring or throw attempt.
EX Fireball, Metsu Hadouken [Corner Only] ( 364 damage, 100 stun )
Use this when pressuring the opponent in the corner. If the EX fireball lands, ultra, if they block it, keep on pressuring. What sometimes has worked for me is to throw a normal fireball, and if they block it, throw an immediate EX fireball. If you are at the right range, they will HAVE to block the 2nd EX fireball. If they try to jump (and they'll be looking to jump to get out of the corner) they'll get hit, which will allow you to
METSU SHORYUKEN COMBOS
Level 2 or 3 Focus Attack, Metsu Shoryuken
With level 2 FA ( 433 damage, 150 stun )
With level 3 FA ( 493 damage, 200 stun )
This will be your main way to use this ultra, other than punishing a highly highly punishable move (like a shotos hard punch Dragon Punch, or Hakan's slide). Use this anytime you can land a focus attack and crumple your opponent. Off of a jump-in (unlikely against good player), on the opponent when the wake up (unsafe), or absorbing an opponents poke and then releasing the focus attack (best idea).
Dragon Punch, FADC First Hit, Metsu Shoryuken
This does bad damage, but hey, if the opportunity arises to use your ultra, take it. Use this in the same scenarios you'd use dragon punch > FADC > Ultra 1. It's the same, just less damaging.