There are three types of mage in this game, "spam" (short cooldowns, more consistent damage), "combo" (unload all skills at once and watch the enemy die) and "burst" (high impact single abilities that don't need to be used together to deal high damage). Ryze is fairly unique in that he's both a "spam" and a "combo" mage, two aspects which rarely combine in champions.
Mages you may like would be:
Cassiopeia, a spam mage. She has very low cooldowns and encourages spamming with her cost-reducing passive. Most of her damage is DoT, with nice burst thrown in with her E, which refreshes if it lands on a poisoned enemy, so even that is spammable. Ult is AoE stun, which is always nice to have.
Lux, a burst mage. Long-ish cooldowns, her offensive moves are all high-impact and 2 provide nice crowd control. Amazing detonating passive, adding yet more burst if you can activate it. Also has a giant pretty death-lazor of doom, which is just lovely.
LeBlanc, a combo mage. She has very high mobility with her double dash, and she can return to each starting point at will, so she's elusive. Individually, her moves are fairly meh (hence not being a burst mage) but when you combine her moves, damage increases vastly, mainly due to the extra spike damage that comes from detonating her Q>R combo.
Brand, a combo/burst mage. His moves each have high impact, but if you can get the combo bonuses to activate, it's always nice. His R is like Ryze's E, but obviously much more powerful. Nice to note is his ability to shred health with his passive.
Finally, Malzahar. He has high cooldowns, but I still think of him as a spam mage, because everything he does focuses on constant damage. Even when he's not casting he's still dishing out damage thanks to his voidlings and long-duration DoTs. Also note that he too can shred health with his W (which is never to be underestimated).
Since Ryze is a dynamic, active mage, you should avoid slow-paced champions like Fiddlesticks, Karthus and Anivia. They feel a little sluggish. --- Evelynn is my waifu. Help... Me...