What skills are you needing to Train? And what gear are you using? Maybe we can help you figure out why you're getting killed so often.
For Archery, the bet way to level it up quickly, especially in the early game, is to use the Bound Bow spell. The Bound Bow, even on it's own, does 18 base damage. That's slightly better than an Ebony Bow, and slightly worse than a Daedric Bow, and this is without the Mystic Binding perk. With Mystic Binding, it's better than the Dragonbone bow! Since XP for Archery is based on the base damage of the weapon, you'll get more XP from a Bound Weapon than you will the normal weapons you can get at the beginning of the game. Now, Bound Bow is an Adept-level spell; with a Conjuration skill of 15, you'll need 183 Magicka to cast the spell. You'll need some Fortify Magicka effects to pull this off, but the overall effect is that you'll get more XP per hit than you will with any other bows. Just note that Sneak Attacks with any weapon give you Sneak XP, so you'll want to find some Bandit dungeons or critters to build your skill up on; just make sure you're not in Sneak Mode so you actually get Archery XP instead of Sneak XP.
For Sneak, your best bet is to get Sneak Attacks. Melee sneak attacks gain significantly more XP than anything else you can do, so they're generally the way to go. However, if you're looking for a safe way to get Sneak XP, you can always use the bucket trick. Get a bucket, then put it on someone's head via the Grab command. Once it's on, find a corner and sneak towards it. You'll get constant Sneak XP because they can't see you (thanks to the bucket), so it's a safe (though cheap) way to boost your Sneak XP.
As for getting the Bound Bow spell, you can get it for free in Fort Amol. You'll have to get to the Fort Amol Prison area, where you can find it in a bucket underneath a Lantern. The Prison can be entered from the courtyard, but be warned that you have a couple of mages outside in the courtyard, and a couple inside the first room of the Fort Amol Prison area. You may want to bring a companion to help you for this.
Also, I'd suggest Muffle. The spell Muffle is an Apprentice level Illusion spell, and it only costs 127 Mana for 3 minutes. This is probably the biggest reason you're getting detected: movement creates noise, which allows enemies to find you. Most of the Muffle effects on pre-enchanted gear have a magnitude of 0.5, so your armor still generates noise as you move; it just generates half as much. The Muffle spell has a magnitude of 1, so it completely silences you, allowing you to move silently. Of course, if you don't want to use the spell, you can always combine a set of pre-enchanted Muffle boots with Sneak's Muffled Movement perk. This perk gives you a permanent 0.5 Muffle effect, so any set of pre-enchanted boots will give you full Muffle.
If you don't have the Unofficial Skyrim Patch, you can make friends with Faendal and take him as a follower, then train with him and take your money back (the USKP eliminates this exploit if you do have it). If you've joined the TG, you can find trainers for both archery and sneak there. I think you can also level sneak by continuously walking into a wall behind a merchant's back: as long as they can't see you and you're moving, the skill will increase. Otherwise, you shouldn't have an issue leveling up those skills naturally: they level pretty quick if you sneak everywhere in dungeons and sneak-kill everything with bows. Draugr crypts are especially good for this, as you can sneak-kill the sleeping draugr before they even wake up, or at least take a big chunk of their health off.
When I can`t be bothered raising Sneak normally for new characters, I just head to Whiterun and go up to the guardtower outside the main gates, the guard always has his back turned, so strip off and face into the corner where both walls meet and plonk an elastic band to hold the controller in a diagonal position and goto bed leaving him to get to 100 sneak for the next morning..
Don`t blame me if your console or PC blows up overnight though!
Also be aware that if you do this, your sneak gains will boost you up a lot of levels but your combat skills will still be low, so you have a tougher time with higher levelled enemies until you can raise those combat skills.
It`s the lazy, quick way to master Stealth
An easy way to build up sneak is to go through draugr ruins while sneaking. stealth kill every sleeping draugr with a power stamina attack.