As a rule, altcoins can be bought for bitcoin. Each new cryptocurrency is trading at the exchange in a couple to bitcoin. You need to buy bitcoin and to bring it to the exchange (or a little) in which listing there are cryptocurrencies which you want to buy.
You can store them at the exchanges, but it will be safer to bring them to the purse. Having come on the official website of cryptocurrencies, you can find information on purses on which it is possible to store it (wallet).