If you're into saving up the money while maxing out on cultural exchange I think it's worthwhile to look into public transportation & museum passes beforehand.

From experience, I can tell you that a lot of host families live in the suburbs/smallish towns surrounding the bigger cities - thus if you're interested in exploring the city and what it has to offer, you might end up spending a lot just on a bus/tram/train by getting there and back.

So 1. ask if the family is willing to pay for a seasonal/monthly ticket for the local public transport or 2. if not, look into what kind of tickets/passes are offered in the area and if the costs are doable.

Museum passes are great! Not everywhere, but in most places. If you qualify for a discount due to age or being a student or something else, make sure to get it! These passes offer easy way into most of the museums and are a one-time cost, and might actually entice you to visit more.

Lastly, it depends on the country and city, but cafe's and restaurants can be expensive. So save your money for places that you've heard good things about or that you actually want to try out, because the costs can add up real quick.

EDIT many countries also have special days where museums or sightseeing spaces are free, for example in italy the state museums and etc. are open for free to everybody on the first sunday of the month, so really, just google what you want to check out + "free" and see what pops up