Don't use your abilities very often early unless you have to push waves because you will run out of mana very quickly. Once you get your Tear of the Goddess you can start using your spells more liberally to push and to harass the enemy laner now that you'll have the mana to do so.

Expect the enemy jungler to try to gank you since Ryze is weak early and not very good at avoiding ganks. For this reason, try not to push your lane, you ideally want to try to freeze your minion wave nearer to your turret than to theirs. Do this by not auto attacking your minion wave or using abilities unless the enemy mid laner does. If the enemy mid laner has good waveclear they will often try to push as fast as they can to force you to farm under your tower and miss CS/gold. If they are pushing hard you'll want to match their push by constantly auto attacking the wave and using your AA+E+Q & E+E+Q waveclear combos. If you do have to farm under your tower you can kill the caster minions easily by auto attacking the first caster minion that get hits by the tower and then using E to last hit it, this will cause flux to spread to the other caster minions making them last hitable by 1 auto attack after the tower hits them.

Although you have a weak early game you will want to abuse melee champions with your auto attacks, especially at level 1 (like Yasuo & Fizz), since they won't be able to retaliate.

Try to always keep a ward in/near one of the bushes in mid, and try to hover towards the side of the lane your ward is in, that way if the enemy jungler comes for a gank on the side you have warded you'll know and you can run away, and if they come for a gank on the side you don't have warded you'll have more time to run away.

Pay attention to the mini-map! If your jungler starts a fight with the enemy jungler you'll want to get to the fight first to swing the tide of the battle in your favor. An early kill on the enemy jungler, or just securing the scuttle, will make a massive difference in the eventual outcome of the game, as small leads early turn into big leads by the end of the game. (I repeat, early kills are more important than a few missed CS in the mid lane!)

Ryze starts to pop off once the mid-game rolls around, especially once he gets his first two core items, Rod of Ages and Archangel's Staff.

Once Ryze gets his first two items he should be able to 1v1 most champions in the game, and because of this Ryze wants to be very aggressive so that he can get kills and snowball out of control.

Push waves and look for fights whenever you can. Ping the lane you want to roam to to let them know you are coming and ult in behind the target/s you want to pick off (even better if you can bring your jungler with you for the gank). Once you & your team get a pick or two push that lane and get a tower and/or go for an objective like a dragon or baron.

Ryze is a good duelist with good waveclear and map control due to his ultimate, this makes him a very good split pusher so don't be afraid to take a side lane, shove hard, and play aggressively to get a kill or tower. Try to maintain good vision in the enemy jungle so that you can run or ult away if the enemy team sends multiple members to gank you for pushing.

Now let's move onto the late game.
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Late game Ryze is a sight to behold. His damage per second is insane and he is very hard to kill for a caster.

Ryze works well as a team fighter or as a split pusher. You can split push or you can group up with your team to fight, it just depends on your team comp, the state of the lanes (where the waves are at and which towers are up/down), and whether or not you have TP.

I'd recommend split pushing bot only if you have TP so that you can join the fight with TP if a team fight breaks out. You can split push top if you have your ultimate up so that you can run away or join a team fight if needed. You can stick with a few teammates mid and push and then use your ultimate to flank and kill their split pushers. Ryze's ult gives himself and his team a lot of options if used correctly.

During big 5v5 & 4v4 team fights Ryze wants to delete enemy squishies, but he can also melt tanks if he has to. Ryze doesn't want to get too close to the enemy team because he is vulnerable to getting deleted if he isn't careful because while he is tanky he isn't a full tank. Because of Ryze's shorter range he's best utilized in front of his back line but behind his front line. That way he can sometimes get within range of their carry's while still being somewhat protected by his own front line. Diving into the enemy front line to get at their carries is a surefire way to get yourself killed before you're able to dish out your amazing damage.

In summary, Ryze has a lot of options in the late game due to how powerful he is individually and how much utility/mobility he brings to the team with his ultimate. You'll want to play a lot of games with Ryze so that you can learn what works best in different situations.

If you really want to improve playing with Ryze, I recommend watch high elo players replays on youtube. Just search: Ryze vs , and than choose the matchup that you want.