You can have a local bike shop do all of your tune-up and repair work. It's not expensive, you rarely need it anyway, and it helps keep your handy local bike shop in business. If you experience trouble while riding, you can just lock up the bike and either walk or call for a ride.
However, fixing your bike is easy once you learn how to do it. That knowledge will let you quickly solve problems as soon as they happen. To get bicycle maintenance instruction, ask a local cycling club or nearby bike shop if they know about a good class.
There are a couple of maintenance needs that you can easily handle yourself and which recur frequently. I discuss these below.
In the US, there are two common types of valves that bike tire inner tubes have. The Schrader (sometimes misspelled Schraeder) valve is the standard type used on all car tires.
Shown below is the Presta valve.
It's best to get a pump that can attach to both types, so that you can use the pump with any bike you might have in the future.
If there is a plastic cap on top of the valve, you have to unscrew and remove it before attaching the pump. For Presta valves, you also have to unscrew the tiny nut at the top, so that it raises up to the valve's tip. Then slide the pump onto the valve as far as it will go. The pump will probably have some way to lock onto the valve. You pump the tire, then undo these steps I just gave.
You will need to own at least one tire pump with a pressure gauge on it. Once per week or so, attach the pump to each tire to see the pressure.
Pump until the gauge reads an amount slightly under the maximum printed on the side of the tire. I have a portable pump for handling flat tires, but normally I prefer to use a floor pump just because it is easier.
If you use a portable pump like the one shown below, you will have to use one hand to hold the pump and wheel at the same time. This will let you pump without wrenching the valve back and forth. Actually, in the photo below, I'm holding onto a spoke; I think holding onto the rim would have been better.
The drive chain, and the toothed gears it touches, are parts of the bike that need frequent attention.
If those parts look dirty, squeak, or fail to travel smoothly, they are probably due for lubrication. Lubrication is best done as part of a whole cleaning process. See my cleaning page for specifics about how to do it.
I suppose you don't have to bother with this, if you own a cheap bike and don't want to make the effort. You could just ride the bike until it doesn't work anymore, and then junk it and buy another cheap one. Maybe you could spray the chain with lube occasionally, when it squeaks. If you don't want to spend time on maintenance, this could be a perfectly valid strategy.
Be sure to check for grit after rain riding, when the tires throw dirty water from the road. You could simply wipe dirt away with a paper towel or cloth, but it's better to do a full cleaning of your bike.