I am new to the game and now have a Shadow Priest at level 60.  She is totally speced with total Shadow Talents.  Is this okay or should I do something different?  I quest alone "All" the time.  Finally, for now, what armor do you recommend for my priest?

Stay shadow til 70 regardless of future plans.
Armor stats you want are Spell damage, Stam, Int and Spirit/mp5.  If you have spirit tap then spirit>>mp5.  Stack spell damage as much as possible, but don't give up massive amounts of the other stats for a small amount of it.

You may want to change to:
http://www.worldofwarcraft.com/info/classes/priest/talents.html?5002300130000000000000000000000000000000000553212210251123051051
putting the points in the disc tree last.  This is a good build for solo/instancing.

That's fine for a build for leveling, providing you're primarily fight mobs at your level or lower.  Shadow Focus is important because it gives you lots of spell hit up to 10%.  If you're fighting mobs at your level, you need 3% hit, at one level higher, you need 4%, for two levels you need 6%.  If he's three levels higher, you need 16%, or 5 points plus an additional 6% from other sources.  You can decide how many points you need there for now depending how what kind of quests you're doing.

Specifically for leveling, you'd also probably want to get Wand Specialization over Unbreakable Will simply because it will let you do some spirit mana regen to finish off an enemy to let you go longer between drinks.

When you're done leveling, definitely make sure you have the Disc Tree as in that build, but get the points out of improved Vampiric Embrace, because VE is nothing but an aggro magnet in most cases, so if you DO put it up, you don't need that much more.  Also, when you're done leveling, Spirit Tap is almost useless in instances and raids, so I'd make sure to take those points out.  Ensure that, if you're raiding, the Spell hit between your gear and Shadow Focus is as close to 16% as you can get it without either wasting the talent point or sacrificing a lot of spell damage.

Shadow Power is also very nice to have for maxing DPS, and is certainly worth having over Silence which is really only useful for the occasional caster pull or PVP.