I need to replace a cassette for CS-HG500-10 but want to go XT

Firstly it's generally a good idea to change chains at around 0.75% wear. This means a cassette will last for several chains, since there is little wear on cassette , chainrings and jockey wheels when the chain is new/not stretched.

However 7400km is not bad.

You don't indicate the state of your other components but normally if a chain goes beyond 1% wear it will have destroyed the cassette and possibly chainrings as well if left long enough

So you will need new chain and cassette. Most chains on eBay and Amazon sellers are fake so don't buy there unless you are very sure about the seller. There really is not too much difference in 10 speed chain durability, but many people prefer KMC for having the quicklink which is easier to fit than Shimano pins.

The XT cassette will not shift better than yours and probably is not more durable (it's Chrome plated rather tha nickel, but given your 7400km your current cassette seems reasonably durable already). The main difference is around 50g of weight thanks to use of aluminium spider arms (the cogs are still steel). 11-36t cassettes aren't as heavy as modern 12 speed cassettes where they go up to 50t or more, meaning that XT has a bigger weight difference.

There is no problem with going to xt cassettes rather than Deore , but not really any real benefit. If the cost difference is small, why not.

To replace the cassette you will need a cassette lockring tool. In my experience the cheap ones on eBay from China for about $3 work just fine. But a branded Park Tool Shimano or similar will be nicer. If you buy the lockring tool you will need either an adjustable wrench or a socket drive. The cassette force is 40Nm which is quite a lot so it's not a bad idea to use a reasonable sized wrench,not the smallest one. OTOH you can overtighten and if you are not mechanically sympathetic you would maybe like to use a torque wrench. These are quite expensive so most people do not bother. IME you get it hand tight plus just a click or two more. If it is not tight enough when fitting you will destroy the freehub body threads over time. If it is too tight then you will possibly destroy the lockring threads and it won't turn properly

You also need a chain whip tool. A $3 Chinese one also works fine, or you can make your own. Fancy tools aren't required here as it's just a bit of chain to stop the freehub body moving

There is a guide here https://www.parktool.com/blog/repair-help/cassette-removal-and-installation

Edit: 11-34 Vs 11-36 is not much difference; check your RD Spec. Some are specced for 11-34 e.g rd-t670. Probably it's better to stick with 11-34 for these. But it will most likely work fine. With your 11-36 you will have to cut the chain slightly longer than you would for 11-34