So I take fairly good care of my chain, rings, and cassette. Today I noticed a little bit of rust on 2-3 links. Does this mean it is the end of my chain?

I've had the bike for a year. The bike is a pass down from 3 generations ('91 Klein Quantum) I do NOT know the last time the chain was replaced, yes it is stretched, yes the entire drivetrain has worn down with the chain, no I do not have enough money to replace the entire drivetrain.

Again,
1. Is this the end for the chain?
2. Is there anything I can do to keep using it?
3. Will a new chain on very old rings and cassette make shifting catastrophic?

A bit of rust here and there is very common, and usually of no concern. The least concern would be for rust at the edges of the plates or on the rollers. Of greater concern would be significant rust in the center of the plates because it can weaken the chain.

But, chains are over built and only strong sprinters or those climbing steep hills need to worry about chain strength.

From the limited description in the post, I'd say, simply oil the chain to prevent more rust, and forget about it.

1. No.

2. Oil it.

3. No.
But a new chain on a worn out cassette will result in a chain that jumps on the worn out cog(s) under pedaling pressure.
A new chain on a lightly worn out chainring will result in a noise as the RD tugs the chain off the bottom of the chainring.
A new chain on a heavily worn out chainring will result in chainsuck as the chain sticks to the chainring and gets pulled up into the underside of the FD.

Better headings for post:

Rust never sleeps
No Rust for the Weary

If you don't want to spend too much then your options are every limited, its never a good idea just to replace one component in a drive system where all others are also worn. So I would soak it clean, there are many special cleaners of you can use diesel etc, turps etc, if you wish: leave it in to soak change the cleaning fluid until its no longer goes black when you stir it, let it hang bone dry, then re-lub it. Depending on the amount of rust it might be worth touching those areas if on the outside links with a brass wire brush before you clean it.

Rust. Lol. Listen,when I went to rehab my girlfriend's old crappy mountain bike which had been sitting in the rain for years just under a tarp, the chain was rusted to ****. All I did was dump a bunch of chain lube on it and work it a bit and it loosened right up. Looks ugly as heck, but works just fine. Not ideal, perhaps, but given how seldom she rides it, it wasn't really worth the effort of a new chain.

So, no, rust is not the end of a chain.

+1 that a bit of surface rust is no problem for your chain. I've rehabbed some chains that were seriously rusty, to the point that some links wouldn't move, and they worked adequately and lasted quite a while.

There are some good tutorials and videos on the internet that will help you figure out just how worn your drivetrain components are. Usually a cassette is good for 2-3 chains, and the chainrings are good for 2-3 cassettes (your mileage may vary). Cassettes are relatively inexpensive if you stick with entry level components. On many cranksets, it is also possible to just replace the most worn chainring (many riders tend to spend the majority of their time in one ring, leaving the other(s) in relatively good shape). Find out just how worn your components are. The age of the bike has little to do with it, miles ridden, style of riding and maintenance have far more to do with wear. A ruler or Park chain stretch gauge will tell you if the chain needs replacement. If the cassette is also seriously worn, I'd consider replacing it at the same time. Unless the bike has a whole lot of hard miles on it, chances are that the chainrings will be in decent shape.

Here... watch this vid.... and chains don't stretch, they wear.

how to check bicycle chain wear - Bing video

and there are vids on inspecting chain rings and freewheels/cassettes, too., too.

a bit of rust is no biggy... a worn out chain can cause damage to other parts... including your personal parts... think about it.