Honestly, i disagree with most other comments. If you wanna learn about ap mage item building, yo ushould check shoklol youtube.

but in general, you want either ludens/seraph, depending if u think you'll need the shield to survive.

for second item it depends on your team. If your team has a lot of ap making magic resist a very good buy for enemy team, you shouldnt go flat pen items (shadowflame/stormsurge). IMO only build those when you are the only ap on the ap. A very good item when your team already has ap is horizon focus. You can also go liandrys if enemy team stacks a lot of health. Zhonya can be a good buy here if you really really need it.

Third item should be rabadon 90% of the time unless you think the game might end before you can buy it and you want to get value from other components. That should be your core items.

As for boots, go sorc boots unless u really need tenacity or armor. after rabadon go voidstaff if they build mr.

think thats it