Yes. You don't need to download the @blockchain to transact on the network. You can connect to a @remote-node, which stores the blockchain for you. All the most common @wallets (including GUI and CLI) allow to use remote nodes to transact on the network. There are multiple ways to take advantage of this functionality. For example GUI and CLI offer a 'bootstrap node' feature, which allow people to download their own blockchain while using a remote node to immediately use the network. Ways to improve the usability of the Monero network are constantly being explored.
You are strongly advised to verify the hashes of the archive you downloaded. This will confirm that the files you downloaded perfectly match the files uploaded by the Monero development workgroup. Please don't underestimate this step, a corrupted archive could result in lost funds.
You can buy Monero from an exchange or from an individual. Exchanges are the most common way to buy Monero; there are compliant exchanges in most jurisdictions. Some wallets include functionality to easily buy Monero with fiat or other cryptocurrencies. Alternatively, you can try mining Monero to get coins from the @block reward.
