It’s better you reach the cities on weekdays because the frequency of public transports inside the city decreases and most of the shops close by 8 pm on Saturday and only open on Monday.
German people generally love to travel during weekends so there are higher chances for you getting a ride. But even if you reach the place on a weekend, all is not lost as fast food joints like McDonald's and Subway are open 24*7 (at the railway stations only).
If you need a place to crash you can register at couchsurfing.org to couchsurf. The members might preferably host you on weekends. Or if you are in a decently big town or city you can use the waiting lounges at the railway stations but the downside is you have to sleep sitting up.
Once you have reached the city it is economical to buy a day pass to use public transportation inside the city. Another suggestion is you can buy a regional eurail pass at germanrailpasses.com/passes-prices that will allow you to travel between distant cities.
I am in Germany right now and if you have any other questions please comment! Hope you have an eventful trip!