ももたろう

「ももたろう」
I am aware that it means 'peach boy'.
My question is about たろう.
What exactly does it mean?

It's a slightly old fashioned boy's name. It doesn't really have a meaning as such. It is either a name by itself or in combination with another syllable like Kentaro, Kotaro etc.

TARO is a name used for the first son of a family and means "the best" :)

I am currently trying to read this story, to improve on my vocabulary. Problem is, I keep comming across things I don't know or can't find in my dictionaries.

しばかり from おじいさんは山へしばかりに
I understand 大きなももが, but the rest is a mystery... 大きなももがどんぶらこどんぶらことながれてきました。
Thanks for the help! :D

しばかり（柴刈り）to gather firewood
どんぶらこどんぶらこ describes the way how the peach is floating on the river
ながれてきました(the peach) was floating (to them)

Lynn,

It looks like you have discovered why children's books written largely in hiragana are not well-suited to language learners.
I would recommend looking for texts aimed at adults with furigana (hiragana readings) over the kanji.

The following free online dictionary covers almost of the words you were looking for: http://jisho.org/