While less weight will help you climb better, you can pay quite a bit to lose a little weight.

Some of the cheaper things to change can be the stem and seat post... but the weight savings can either be significant or minimal depending on what the current components weigh. A relatively light handlebar can also be pretty inexpensive.

The best performance upgrades according to many are wheels and tires, and the argument is that rolling weight is more effective... Wheels and tires can be had for varying price points, and can save quite a bit of weight, depending on how much you care to spend. Just going to narrower/lighter tires and tubes can help, but you could become more prone to flats too...

If you carry a water bottle, you would probably save as much weight as any upgrade by not carrying the water bottle... but you obviously lose out on your ability to stay hydrated.

You can also cut down on how much stuff you carry on your back.