Find a project that you really believe it will rise in the long term. Check out the market cap, the project team, roadmap, current state of the development, and worth of the market they are trying to cover. What are the problems they are trying to solve, and who are the competitors. Doing proper research will make you feel more confident when holding. Check also if the coin has fixes total supply and what percentage of the total supply is in circulation to predict future inflation. You can find a lot of useful information about the coin on
https://coinmarketcap.com/ . If you click on your desired coin under tab/markets you can see list alt-exchanges where you can buy it. Another useful website is
http://coinmarketcal.com/ where you can find dates of interest for various coins.
Most of the alt coins are trading on exchanges with altcoin ->BTC/ETH pairs. So first you need to obtain BTC or ETH. Coinbase is the most popular choice for that purpose but I would suggest you
https://www.bitstamp.net since they have lower fees (0.25% compared to 1.5%). Another option to lower down fees is to use GDAX from Coinbase. With GDAX, if you're "taking" an order (meaning filling an existing order in the order book), the fee is 0.25% for Bitcoin and 0.30% for Ethereum. However, maker fees (meaning placing a limit order on the order book that may or may not be filled) are 0. After you bought BTC you can transfer it into BTC wallet on alt exchanges and start trading. Some of the popular alt exchanges are Binance, Bittrex, Bitfinex, etc. I prefer binance since they have the lowest fees (0.1% compared to 0.25%/0.2%). I also grew quite fond of their user interface. Check out advanced mode. You can use this referral link if you like
https://www.binance.com/?ref=10196951 I would also recommend to use 2step verification for logging in.
Don't buy at all-time high if the price was rising too suddenly. I usually search for the dip for the coins that I believe in long term. Rule of thumb: all cryptos are pretty much influenced by BTC. When BTC rises suddenly most of them fall. When BTC falls down suddenly most of the cryptos fall. When BTC is stagnant or has a slow steady rise, alt coin flourish. And last, remember to take part of your profits when it's skyrocketing. Usually, there is a correction after sudden increase. Just holding, without taking the profit is still haunting me with my NEO (Antshares). It's always good to take profit to increase your stack. Have fun and good luck trading/investing!