Of course, you could be a khajiit mage for all I care!
For archery, you're best bet is a Wood Elf and for mage, a High Elf...
A while ago I played a character who used light armour, archery and conjuration magic. I quite enjoyed that build, but had to retire the character early, as he was using archery a lot more than conjuration, and that conflicted with his backstory.
Also, my current Daggerfall character uses archery for ranged combat, and Destruction magic for melee. But that's Daggerfall.
By the way, it's good to see someone who can spell "favourite" correctly.
My old Ranger had Archery, Stealth, and Alteration. Some Conjuration, ad I was working on Illusion as well for silent casting
Light armor, bound bow, ice enchanted sword, frost atronach.
Light Armor
Sneak
Archery
Conjuration
Illusion
Throw Fury(or a few) into a Crowd, then Conjure the Bound Bow and start picking them off.
Works efficiently for groups of 2 and Up. Single targets are just easily picked off.
Usually, unless something went wrong, there will be 1 Survivor with depleted HP. Easy enough to pick off with a Sneak Attack w/ a bow.
Good Times.
Why not just play as a Dark Elf and use pretty much everything? I mean... Dark Elves are known for their skill with blade, bow and destruction magic, so they're good with each offensive group. It's not difficult to just extend that to every other group too. Put your Dark Elf in light armour, sneak around a lot and use a bow to initiate combat with a sneak attack, then switch to a one-handed blade and use magic in your other hand. Use every school of magic. Use destruction, conjuration, alteration, restoration, illusion... use them all.
There you have a character that is an archer, a warrior, a mage, can be adapted to literally any situation AND never gets boring because if you get bored of using the bow, retire the bow for a few levels and become a spellsword. If you get bored of the bow and the sword then become a pure mage (complete with robes) for a few levels. It's a great character to play... especially for someone like me who gets bored of a character around level 10 and wants to start afresh as a heavy knight or something.
Get your archery up and conjuration, what I do is I have a frost attronach in one hand. Then I conjure the attronach then I quickly conjure my bound bow to give him some long range support