If you're looking to buy and start trading actual cryptocurrencies and not CFDs on Cryptos - Tickmill is not the place.
You should be looking for an exchange like hitbtc, gemini, bittrex, poloniex, coinbase or one of the other big exchanges.
The main issue is to find one that is big enough and not a scam. The ones I've listed are some of the biggest and most diverse exchanges.
As far as regulation goes - at the moment there are no actual regulatory authority and regulations for crypto exchanges, so you must be extra carefull when deciding where to start trading.