私が全部負担しますよ。どうですか。あなたのおかげで、私の気分が転換できて、何か恩返ししたいんです。せっかく同じ地元じゃないですか。
私が全部負担しますよ。==> I will bare the cost.
Yep that's correct. 負担means to bare. The object can be cost, load, effort, etc... Here it seems to be cost (of the trip)
どうですか。==> How is it?
A more natural translation might be "How about it?", where "it" refers to his proposal of paying the other person's cost of going back to Hokkaido.
あなたのおかげで、私の気分が転換できて ===> As a thanks to you, I will share your burden????
私の気分が転換できて:a too-literal translation is "My mood was able to be changed", but a more natural translation might be something like "I'm feeling a lot better", so the total sentence becomes "Thanks to you, I'm feeling a lot better". A very common expression is 気分転換(きぶんてんかん)which means a change of pace, a refreshing change. Check it in Eijiro
http://eow.alc.co.jp/
何か恩返ししたいんです。==> I don't understand this part, please help me.
恩返し(おんがえし)is the paying back of a favor or kindness so this means "I'd like to do something to pay you back" Look up 恩返しin the dictionary if you need more detail or in Eijiro if you need more example sentences.
せっかく同じ地元じゃないですか。 ==> I also don't quite understand this part.
He is saying "we are from the same place, aren't we. (so why not visit home together?)" 地元(じもと)in this case means home town, local area. A very common expression is 地元の人々(ひとびと):local people, the locals
Hope this helps!