I have a question.

Should I alternate raising health and stamina, or just stick with health? Are there any real benefits from raising stamina besides raising my carry weight?

Depends. Stamina means more power attacks, bashes, zoomed in shots with the bow, sprinting, and carry weight.

Do you want these fabulous prizes?

Well it depends. What type of character are you?

Well it also raises your stamina for power attacks, zooming, sprinting, and the like.

Depends on your build. Mages probably wouldn't need stamina all that much. Archers and sword'n'board warrior types depend heavily on it, though.

As a rule, I try to get 200 base health/stamina/magic then work from there. That might nerf your character badly if you're playing on expert/master though.

Stamina is very useful for melee. More power attacks and shield bashes! Also if you're an archer, it allows you to zoom in for longer (assuming you have the perk.) Personally I alternated. Not sure what the best way to do it is but this worked well for me.

The carryweight boost helps, but other than that it really depends on what kind of character you're playing. Archer's don't need stamina, mages don't need stamina, dual wielders, two-handers, and shield users need a lot. Health becomes a lot less important as your character gets better at smithing - damage resistance leads to better durability, meaning that at the armor cap of 80% DR, you're only seeing 1/5th the damage, and therefore your health bar is effectively 5 times as big. There's nothing that will give an apparent boost like that to your stamina. So early in the game, when your armor svcks, boost health mostly. Once you notice you're not dying so much, stick to stamina, provided your character needs it.

Not sure if mentioned, by Stamina also lets you increase your carrying weight.

I left my stam at 150 and got the perk from the restoration tree that restores stam when you heal yourself. Basically gives you limitless stamina as one cast of heal will give me almost full stamina.

It depends on the type of character build you are playing really.
But at low levels being able to sprint away from some of the higher level enemies can help any character out of a jam.
I'm currently playing a light armor shield and war axe character,I'm going with 60% health and 40% stamina.
Haveing enough stamina to keep shield bashing during an extended fight is great.

If Mana is not an issue, either because you don't use a lot of magic or none at all, then a good rule of thumb is to put 2 Levels into Health for each one you put into Stamina. Stamina is important in fights, but Health is even more so for survival.

Exactly what I do with a melee character.

Honestly, I usually set my encumbrance to 100 or 150 at the start of a character, so I usually go even between stamina and HP.

My current charecter is a mage and I find Stamina to be very important to me since I like to fight with my dual conjured swords. I play on master level and just throwing fire, ice or lightning bolts at my enemies isn't good enough.

if you want a real challenge keep your health at 100 and play on master, you know i might try this for my next build, i think it would make the game more realistic.

3 health, 2 stamina for my sword'n'board Redguard and my dual-wield Redguard.
1 health, 1 stamina for my greatsword Orc.
3 health, 1 stamina, 1 magicka for my stealth dagger/bow Khajiit. (Haven't gotten any spells to need the extra magicka tho. . . .all well)

As above, along wiht Carry Weight, Stamina also affects power atacks, blocking, sprinting and zooming in with a bow.

My current character is a 2-hand, light armor, melee warrior, level 40. I put everything into health. It has been a very easy build to play.

Well I'm level 30 atm.

I use a bow until they get close, then switch to a sword and shield. The only mage skill I use is restoration so I don't need much magicka at all. I raised it once before realizing this so it's at 110.

SInce then I've alternated between health and stamina, both are at 240. Carry weight only goes up by 5 when I pick stamina so I'm not too worried about that, but if it's that essential to archery, block and one-handed, I'll keep alternating.

My sneak Archer has a similar style, once my stamina hits 250-300, then i am going to concentrate on health.

edit: i'm also trying for the Pickpocket perk that allows an additional 100 carry weight(and the place poisons).

here's a challenge: put all your perks into magic, divide the magic/health/stamina points however you feel and never use magic. be a fighter or thief like character.

600 health, 200 magicka, 200 stamina.

You can regenerate stamina via absorb stamina on your weapons, vegetable soup or respite.

Most expert spells cost just less than 200 magicka when perked and at 100 skill. Having 200 magicka lets you cast the most spells without sacrificing too much.

stamina is currently bugged,(will probably be fixed in a soon to come 1.4) and lets you do even the most complex moves as long as there is 1 stamina available. With that in mind, judging the real amount of needed stamina is kind of difficult.

That's a bug how? The only problem I see is that there is no reverse stamina bar for overdoing actions that require stamina.

The bug is that if you consume something with a stamina regen bonus (like chicken soup) or use absorb stamina enchanted weapons, you will always have at least one stamina, and will always be able to power attack. Compare this to magic where running out of magicka means not being able to cast spells, even if you've taken a magicka regeneration potion.