I bought a battleaxe in the town(first screen rightmost shop) but I assume that there are better ways to get an axe.
To get a free level 1 axe you need to Talk to Bianca (Girl in the right mansion in the third part of the town) after the second day.
Get the free level 1 axe from the rich girl in the mansion, then you can upgrae your axe. When it's fully upgraded, you can even cut a wood stump in one hit!
rune factory is quite fun.... its like harvest moon with added action..... there is this walkthrough website that may be of help if you need it.... http://fogu.com/hm/rune_factory/
this game is using standard role system.... so, it means after u defeat the first boss, u'll get the pass to the next dungeon.. as well as u have already completed a special current state.... to get the access to the second cave, after u defeat the first boss(greater demon),u need to plow 100 square of the fields inside the first cave(i forgot the name)~.~ then, go to the chief of the village(Godwin).. he'll give u the pass, so that u can advance into the next cave.. (=^.^=)