I want to buy some ico tokens from other users. However I don't have a wallet for that specific type of token(e.g. BasicAttentionToken). Can I do that?
Any token built on the Ethereum platform can be kept on any generic Ethereum wallet. Ethereum-based tokens are regular Ethereum smart contracts so you control them with your Ethereum account by sending Ethereum transactions.
It is pretty simple, a wallet like metamask for instance has a token section. You can simply define the token you want to add inserting the contract address and other few information required by the form and that's all. Metamask will show the balance for all the tokens you own and watch.
Metamask is a simple Ethereum wallet that comes as Chrome extension, but other wallets provide pretty much the same functions.
See Metamask https://metamask.io/