What parts needed with new chainring? I've read that the chainring is that spikey gear wheel at the bike's pedals that the chain egages with. 

The teeth on that chainring are so worn on my bike that the chain jumps off depite adjusting the derailleur.

The teeth of the chainring at the back wheel don't seem as worn, although this answer says they wear faster.

When replacing the chainring, it seems I should purchase:


Crankset
Chain
Chain adjuster tool


Are there other parts that should be changed at the same time?

Here are three photos of different parts of the chainring:
If the chain rings are badly worn, its almost certain the cassette (rear cogs) and chain need replacing. do not change the chain rings on their own. 

The chain rings on most cranks can be replaced. Some cheaper ones (mostly kids bikes) use rivets, and you need to replace the whole crank set. If possible, the easiest thing to do is replace chain rings with identically sized ones. The BCD (Bolt circle diameter) must match, you can vary the number  of teeth but that changes the gearing and might need to adjust or (worst case) replace the front dérailleur, and may also need to replace the rear derailuer. 

If you decide to changing cranks, you will need to match the bottom bracket you have, or get a new bottom bracket. if you need to change BB, you can end up in BB standards hell sooner than you can imagine. Keeping the same BB and Crank set is by far the easiest.

Rear cassette - easiest is replace like for like - same speeds and number of teeth on the big cog. As with the chain rings, if you change the number of teeth you may need to adjust or replace the rear derailleur. 

Chain - if you keep all the gears the same number of teeth, you can make the new chain the same length as the old one (presuming old was correct), otherwise you will need to work out the correct length for the new chain. 

Tools required: 


Chain whip
cassette removal tool. 
Hex keys 
maybe a chain ring bolt spanner  
Crank removal tool for your cranks (usually square
taper, but a dozen others exist) 
Maybe a BB spanner if replacing BB. 
Chain breaker.