First, create an Avalanche Wallet here.
You can then send your purchased AVAX from an exchange to your Avalanche (X-Chain) wallet address starting with X-avax….
Once your wallet is funded, click on the “Cross Chain” tab on the left hand side.
Select which chain you would like to send your AVAX to. Each chain on Avalanche has its own functionality and rulesets, which you can review here.
If for any reason, your transaction gets stuck, navigate to the Advanced tab and you can import your stuck transaction into the correct place.