By reason of the high damage spells and short CD, Ryze is the AP and CC in a team. He is recommended for newbie, because his abilities are too simple. But he is not only used for newbie, because he has potential to be further tapped.

Ignite prevent enemy taking Health Potion in battle. Flash help Ryze run away or chase if he have a chance.They are necessary summoner spells for AP or AD.

Generally speaking,a champion who is not used well by us but gets the right items can not be too bad.So the right items are important for every champion.Of course right item is not the only one to choose,but basically we should know the types of AD,AP and support. Now we know Ryze needs AP items.Then according to the need for mana from Ryze's abilities,we can pay attention to the one that add mana,like Rod of Ages,Archangel's Staff,Frozen Heart etc.

Level1-6：
When we go out on level 1,we can take Sapphire Crystal*1 and Health Potion*3 in order to buy Catalyst the Protector as soon as possible.
When we get level 6,at least we have enough money to buy Catalyst the Protector ,Tear of the Goddess and Boots of Speed.

Level7-12:
Now we have been a little tank and have enough mana to cast ceaselessly,meanwhile the mana bring much damage for Ryze's abilities.We can clear up minions quickly and have spare time to help jungling to clear neutrals or gank.But farming may still be better than pulling the tower.During this stage,we should have Rod of Ages, Archangel's Staff and Sorcerer's Shoes.

Level13-16:
Rod of Ages provides 450 Health,450 mana and 60 AP.So it is a useful item for AP especially the one who need to stay in the fight as long as possible.Archangel's Staff provides enough mana and enhances Ryze's damage.Frozen Heart provides CD, in addition,it provides armor to resist enemy AD. Banshee's Veil provides not only Health and mana,but also Magic resist to resist enemy AP.At last,Void Staff provides Magic damage ignores .At this time or Iceborn Gauntlet's spellblade provides a control to slow enemy.

I see so many players said that he need equip Muramana.But its UNIQUE Passive Awe is same as Archangel's Staff,so the 750 mana can not be added that is tested by me.So the Muramana is not so good.

In general if we want to learn how to play a champion,we should start from his ability. His abilities are Overload,Rune Prison,Spell Flux,Desperate Power and Arcane Mastery.From the presentation of his abilities we can simply summarized in two key points.One point is that the output of his 3 abilities based upon his maximum mana.The other point is that his passive ability can reduce all spells CD by casting a spell.It means that maximum mana and CD are important to Ryze.
Every ability has its function and the detail is as follow:

Arcane Mastery: His passive ability.We make use of it to reduce other spells' CD.
Overload-Q: It's a single target spell and CD is only 3.5 second.So the spell is often used to consume enemy's HP in the lane.
Rune Prison-W: It's Ryze's CC spell.It is used to hold a champion and then we can gank him.
Spell Flux-E: It's AOE spell and used to clear up minions.
Desperate Power-R: It's ultimate spell.Compared with other ultimate spell,the spell vamp is very common,but the short CD is unique.It means Ryze can cast R more than twice in a team fight.And it is absolutely nightmare for tank when the spell work with E,especially all the enemies who stand together will get damage.

Masteries help Ryze cast more abilities and output more.So mainly focus on reducing CD,adding Mana and AP.

I take the Mana Runes to work with Ryze's abilities.Because Ryze is a little weak in early levels,so fixed Mana can provide a certain output.And because Ryze just need standing there to output,so he don't need movement speed.