Trueing a wheel is something you can do on your own, but until you get good at it
it'll take you time to do it right - "carefully tighten the spokes on the opposite side...
" is kind of an oversimplification.
Typically a trueing stand is used; the wheel is placed in the stand and prongs or arms on the stand
determine exactly where the wheel is out of true.
The problem is that sometimes you need to loosen some spokes to get enough play to pull back against the deformation enough to straighten it out.
Someone who does it all day makes it look easy - you bring your wheel in to a good bike shop and usually it takes longer for you to describe the pothole you whacked into than to actually true the wheel.
If you've got a good relationship with them they may not even charge you for a 10 second job (karma would demand an offering of beer or pizza as appropriate).
If you don't have a trueing stand
but you're feeling adventurous you could try doing it with the wheel in the frame - stand behind / in front of the wheel
(so you're looking at the tread) and spin the wheel slowly - you should see where it comes out of true.
Use an actual spoke wrench - if you use pliers you can round the ends of the spoke over and if you use a wrench you'll put too much torque on it
(I have a friend who has done it with a leatherman in the middle of a ride
but he also repaired and managed a bicycle store for many years).
Since it's an inexpensive wheel if you're confident enough in your abilities I'd say go for it - worst case
is you end up replacing the rim, but you may learn a new skill in t he process.
bicycles.stackexchange.com/users/2362/lawndartcatcher
You can attempt it yourself, or get it done at a bike shop.
I'm not sure what bike shops charge these days for truing, but it's probably less than half of what a new inexpensive wheel would cost (though you may be able to get a used wheel cheaper).
If you have moderate mechanical abilities you can attempt it yourself.
The important thing is to get the RIGHT spoke wrench for the spoke nipples, and get a good quality one.
And if the nipples are at all corroded, be prepared to "break free" a nipple with Vice Grips before you end up rounding off the nipple with the spoke wrench.
(If you round off a nipple they can be replaced, but it's a minor PITA.)
(If the nipples are badly corroded give the screw threads of each a shot of Liquid Wrench before you begin, being careful to not let the stuff dribble down into the tire.)
Truing a wheel is mainly a matter of THINKING about which way each spoke pulls.
And you may need to loosen some and tighten others (though you should probably tighten more than you loosen overall).
bicycles.stackexchange.com/users/1584/Daniel R Hicks
As other contributors have noted, this is DiY with a spoke wrench.
Spoke wrenches come in different sizes, take the wheel into a shop so you can get the right wrench.
Deflate the tyre somewhat before you begin as a hard tyre can puncture if making spoke adjustments.
As for the truing process, I have to recommend Ken Kifer's guide because he wrote lots of great stuff and this article serves as an intro to his site: http://www.kenkifer.com/bikepages/skills/spokes.htm
If you do want to get a new back wheel there is the small matter of the sprockets that will need changing over.
Get the shop to do this for you when you buy the wheel as you don't want to be buying special tools for something the bike shop guys do every day.
Also expect a 24" wheel to be special order, but not that expensive if you do get a basic steel hub
/alloy rim model.
bicycles.stackexchange.com/users/1487/ʍǝɥʇɐɯ
I had exactly the same problem a week ago.
My son had a minor crash and the front wheel was slightly buckled.
Brought the bike to bike shop to see if they could fix it and
they said no - and to get a new wheel - cost ~ €40 thank you very much.
Mind you, it's an old crappy bike
so not that surprised that they said no.
My solution - looked around for second hand bike which we got for less than the price of new wheel!
bicycles.stackexchange.com/users/402/Catch22
Of course they can!
It's easy enough and much cheaper to do it yourself with a spoke key.
bicycles.stackexchange.com/users/2358/cmannett85
My solution was to replace the wheel with an undamaged one salvaged from another cheap bike, because the wheel was too damaged to repair.
A tree fell on my cruiser bike and bent the coaster brake rear rim irrepairably.
I looked online and found a new one for about 60 dollars plus shipping.
So I looked around on the local swap and shop forums and saw a few bikes for around the same price.
But yesterday I found a cheap old cruiser bike at a thrift store that also had a lot of other usable parts as well as two fairly true rims with good tires.
bicycles.stackexchange.com/users/46061/Tim
My son has a relatively cheap kids mountain bike, no suspension or anything flash, and after a minor crash into a friend the back wheel is slightly out of true. It only causes issues under braking when the warped area of the wheel leads to stuttering braking, and locking up entirely when the brakes are pulled hard.  My gut feel is that I should just throw it and get a new one, as I don't know how repairable these things are.  Can bike shops repair that sort of thing?
For a slightly warped wheel, is it worth getting it repaired?
bicycles.stackexchange.com
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bicycles.stackexchange.com/users/2077/Rory Alsop
It is probably a good idea to attempt to fix a slightly warped wheel. A bike shop may also do it for a reasonable price and cheaper than a new wheel. A spoke wrench is required to do it yourself and you would need to take the wheel to the shop to get the right size.  Another  solution would be to replace the wheel with an undamaged one salvaged from another cheap bike, if the wheel is too damaged to repair.   This is a job you can do yourself with some skill, or a shop may not charge much. You could try doing it and bring it to a shop if you fail, or just replace the wheel with one from another bike. Make sure you use the proper tools with the correct size, and deflate the tube beforehand to prevent flats. 
You can take it to a shop or try doing it yourself with a proper spoke key. It is not expensive at a bike shop, but doing it yourself can teach you new skills. If you fail, you can just take it to the shop or throw it out. Make sure you use a proper spoke wrench with the correct size, release the nipples and deflate the tube first. It may be easiest to get a replacement wheel from a different bike.