When you're away from home and you want to see a variety of things, you can do two things:
Having a base has the main advantage in that it doesn't waste a lot of time finding good/cheap accommodation and moving in and out every day or two. It can also turn out a lot cheaper as multi-night stays are frequently on offer at significantly lower rates than for one or two nights.
This concept is akin to the military one of a fire-base; an enforced position from which you venture forth into surrounding enemy territory, attacking various targets. [Let me know when I come up with a truly original idea!]
A compass may be useful for drawing circles of equal distance from points of interest on the map. Or you can use a digital map of an image in a graphics package on your computer and overlay it with circles of a suitable size(s). The circle size(s) should be based on how far you think you can travel in an acceptable time; say half an hour or an hour.
Don't be too concerned if some locations are nowhere near prospective bases. Consider using them as points where you break up longer journeys between bases.
You may have to consider several base options between a particular set of nearby targets, balancing costs, facilities and time. Different areas of especially larger cities can have significantly-different costs. It is often much cheaper to stay on the outskirts of larger cities than within spitting distance of town hall, a major railway station or an airport.
Once you've found a place, look around for other things in the vicinity that may also be of interest to you. These will help to fill your days and prevent aimless wanderings that result in you finding something interesting; with only minutes to spare!
Nearby friends and relatives will usually be the cheapest option in monetary terms. Sometimes you can be nicer to them, and yourself, by not becoming a burden by an overly-long stay. They'll often be happier to see you for an afternoon or evening; than for a fortnight! After all; they also have their own lives.
If you take up their generous offer of a roof over your head, then get them involved in your adventure. It gives them an excuse to see things that they might not otherwise.
B&B tends to suit the single traveller, staying a few nights. Holiday homes tend to suit multiple travellers travelling together and staying at least 3 nights in the same place.