The most important enchantments imo for a mage is fortify skill (whatever magic skill you feel is most important), I fortify destruction to around 91% (fortifying a magic skill means it costs less; in my case 91% less than the base cost) which means that I can throw thunderbolt (expert lvl) without there being any serious risk of depleting my magicka reserves (it costs me 9 magicka per cast, which is lower than the base cost for Flames) – basically I’m an unstopable destruction mage (even the master lvl spells have a negligible cost). Secondly magicka regeneration should be prioritized and probably more than fortifying your magicka reserves – the faster your magicka regenerates the faster you can cast new spells, when you run out of magicka. Thirdly fortifying your magicka reserves are also something to consider, but don’t go overboard (as I probably did a little myself) as magicka regeneration is more important in terms of enchanting (you can always increase your magicka reserves when lvling up). I would generally recommend fortifying destruction as most other spells can be cast when needed for the magicka reserves you build up through lvling up (+ investing in novice etc. perks will reduce cast cost) and rarely needs to be cast as much as destruction spells. Keep in mind that there are limitations to what enchantments can be used on the specific parts of your equipment (fx you can’t enchant your gloves with all the samme effects as your ring).
The perks needed in the enchantment skill-tree (for mages it's the only enchanting perks you need, but it might be different for other class types) are: Enchanter (all possible upgrades for this perk are necessary for full optimization), Insightfull Enchanter, Corpus Enchanter and Extra Effect.
My equipment is as follows: Ring with fortify destruction 26% and fortify magicka regeneration 65%; Circlet with +62 magicka and fortify destruction 25%; Boots with 37% fire and frost resistance; Gloves with +62 magicka and fortify lockpicking 40%; Archmage’s Robe (the one you get awarded with at the end of the College of Winterhold questline) with all spells fortified by 15%, + 50 magicka, fortify magicka regeneration 100%; Amulet with +62 magicka and fortify destruction 25%. To add it all up: fortify destruction 91%, fortify magicka +236 (in retrospect i should have prioritized regeneration more than the actual magicka pool – but hey I can always enchant new equipment), fortify magicka regeneration 165%, fortify lockpicking 40%.
All in all my mage character (currently lvl 47) is very powerfull and the talk about magic users being underpowered at higher lvls in Skyrim is something I simply can’t recognize at all.
For stealthy characters I guess fortifying skills like pickpocket, lockpicking, sneak and maybe archery and one-handed to increase damage would be good choices; also fortifying stamina will be a very good choice (and maybe a little health increase). Also some side equipment with alchemy fortified to use when making potions and poisons would probably be a good idea.