How do you replace the transfer case on a 2010 Toyota RAV4?
Research availability of new transfer case for this model... usually transfer cases are too specialised for generic parts shops, so you need to go to a dealer.
If the price is too high from the dealer, you could get a second-hand unit from a junk yard or dismantler.
Once you've sourced a replacement transfer case, you need to remove the faulty one.
Lift the vehicle safely using a hydraulic jack and jack stands or a car lift.
Remove the driveshaft from the transfer case.
Identify which bolts are connecting the transfer case to the subframe and/or transmission.
Remove the mounting bolts of the transfer case.
Separate the transfer case from the transmission.
If the transfer case shares fluid with the transmission, you may new to replace a gasket or use a silicone gasket material on the mating surface.
Install the new transfer case.
Reconnect the driveshaft.
Depending on the design it will use either it's own fluid or shared with the transmission. Top up any missing fluid.
Lower the vehicle from the lift or jack stands.
Test drive the vehicle.
Find out which generation your car is in.
Generally, cars go 4-12 model years between major re-designs. Take the Pontiac Firebird: "Smokey and the Bandit" was 2nd generation (1970-1991) and "Knight Rider" was 3rd generation (1982-1992).
Within a generation, most things will bolt and fit the same. (although the parts may not be the same). Between generations, don't get your hopes up. All that to say, don't practice on the wrong generation.
Model years run half a year ahead, so go by model year not calendar year. A 2010 RAV4 seems to be 3rd Generation, which is model years 2006-2012. *That's great news. 2006 cars will be plentiful at picking yards.
Research the work ahead.
You either use Web resources (Youtube videos, RAV4 forums) or you get the service manuals from Haynes or preferably the shop manual from Toyota.
Meet "Pick Your Part" yards.
These are wrecking yards open to the public where you pull the part. Each car is neatly laid out in an open field, propped up on hardy jack stands. You bring your own tools and wrench the parts off yourself. The part prices are laughably cheap (most of the time). There is a small entrance fee, but that is a good thing - that keeps tool thieves away. A local pick-your-part was closing and had free admission their last week, complete disaster.
The beauty of pick-your-part yards is you can practice for free (well, admission fee) on a junker that nobody cares about. That lets you learn what you're doing non-destructively, and discover what tools you need and don't have. Without risking your own car. Since their actual goal is to sell you parts, it's also a part source.