A swap asks for a destination address before it will do anything. If you do not already have a Monero wallet, that prompt arrives at the worst possible moment: you have decided to act, and now you are picking custody software in a hurry. Choose first, swap second.
The one rule that matters
Whatever you pick, you must control the seed phrase. A wallet that does not show you a recovery seed is not your wallet. An address someone else holds the keys to is their money that they let you look at.
Write the seed on paper. Do not photograph it, do not store it in a notes app, do not email it to yourself. Anyone who reads it owns the funds, forever, with no recourse.
The official full node wallet
The GUI wallet from the Monero project syncs the entire chain to your machine. It verifies everything itself and asks no third party anything, which is the strongest privacy position available.
The cost is real: tens of gigabytes of disk and a first sync that takes hours. Worth it if you are holding meaningfully and can leave a machine running. Overkill for a first small swap.
Light wallets
A light wallet keeps your keys locally but asks a remote node to scan the chain for you. You keep custody; you give a node operator a view of which transactions concern you.
This is the sensible middle for most people, and it is what most first-time users should use. Prefer a wallet that lets you choose the node, and change it away from the default if you care about who is doing the scanning.
Mobile wallets
Convenient and fine for spending amounts. A phone is a general-purpose device that installs software constantly and gets lost and stolen; treat a mobile wallet as a pocket, not a vault.
Hardware wallets
Both major hardware vendors support Monero. Keys never leave the device, and a compromised computer still cannot spend your funds. This is the right answer for meaningful amounts held over time.
Setup is slower and you need the device present to spend. That friction is the point.
Before you paste the address
Check the first and last characters against your wallet after pasting. Clipboard-hijacking malware exists precisely because people skip this step, and a Monero transaction cannot be reversed or recalled.
Send a small amount first if you are unsure. The fee on a test transaction is trivial compared with discovering a mistake at full size.