Taking the first option listed above, which is to buy the underlying, you become the direct holder of the digital asset. Upon purchase, the cryptocurrency is sent to your bitcoin address or account (wallet) with the exchange. From there, you can transfer the crypotocurrency to any bitcoin address or wallet address using your private key that verifies you control ownership of the asset.
Always pay attention to Bitcoin. Most altcoins (every cryptocurrency except Bitcoin) are pegged more closely to Bitcoin than Asian currencies were to the USD during the Asian Financial Crisis. If Bitcoin price pump drastically, altcoins price can go down as people try to exit altcoins to ride the BTC profits; inversely, if Bitcoin prices dump drastically, altcoin prices can go down, too, as people exit altcoins to exchange back into fiat. The best times for altcoin growth appear when Bitcoin shows organic growth or decline, or remains stagnant in price.