Backpacking through Europe

I would like to go backpacking through Europe the day I graduate, not literally but you get the picture. I have two options for backpacking through europe. I can save up through the year (which would be totally difficult because I hardly make enough to save) or wait until July 13 when I am legal to recieve my ‘trust fund’ (believe me, not much there, but enough for college and a few expenses like backpacking). If I do wait that’s only probably a good month of traveling where if I leave in June I can get another twenty days of traveling. Twenty days is a lot. I wish I would be able to have the bank account opened while I leave but apparently I have to sign papers on that day before it is open. Is there any legal advice? Is a month of traveling okay or the twenty days valuable?