This expedition relies on your town having public transport. If you don’t even have a few buses, maybe you could spend the day lobbying your local MP for better public services instead.
If you do have transport, buy yourself a day pass and get on whatever bit of it goes closest to your home (if you have to drive somewhere more central and park the car, then go right ahead and do that). Stay on the bus, tram or whatever until it gets to the end of the line, or you see something you want to have a look at, or you get to a spot where you intersect with another route. Get off, have a look around, take some photos and pick up a few more suburban souvenirs, go into a strange cafe, shop or bar, then hop on the next bus that comes along and keep going. Continue until it’s dark, or you’re too bored to continue.
Try not to deliberately choose destinations. If you have a lot of buses, trains or trams to choose from, don’t read their destination sign: just choose the first that comes along. And remember to keep your ticket! You’ll want a souvenir.