According to the 2012 results of This Poll
, Vegas or anything on that list should be fine.
It should be noted, as mirkosp said, that Sony Movie Studio
(~$50-$100) is *NOT* Sony Vegas Pro
(~$600) <---(that extra zero is not a typo)
The differences are in the features. Like Pro has Masktools to cut out characters frame-by-frame; IIRC, Studio does not.
More differences can be found here:http://www.sonycreativesoftware.com/vegassoftware
However, masking can always be done manually using PhotoShop or GIMP
Now, if effects are not that important to you, almost any program will do.
I personally use Magix because I consider it to be a budget editor (~$50-$100).
There are tutorials on the page above explaining how to use it.
Of course, according to the poll, most AMVers use some flavor of Sony Vegas, so you may find more tutorials for Vegas online; albiet, probably for Vegas Pro... despite costing ~$600. Same with After Effects... despite costing $1250.