The procedure for investing in foreign stock differs from country to country. To invest in the US markets, you can either go through any of the leading domestic brokers who have tie-ups with US brokerage firms or sign up with an online US broker directly. Your decision will be a function of the quantum of your investable surplus, cost, convenience, and frequency of deals. 
For small and infrequent deals it may make sense to go through a domestic broker. Brokerage, administration charges, and other fees tend to be higher in abroad. In most European countries, you usually have to open an account with a broker in the relevant country before you can trade.

It is very easy for you to start trading inequities abroad. You need a bank account with a branch that allows foreign remittances, and an account with a domestic broker. They have made it very simple to open your overseas trading account with their partner (foreign) brokers. These International equity brokers allow you to use their platform for trading.

For investment, you have to transfer your investment corpus to your brokerage account by filling up Form A2. The money is transferred in a day or two. You can buy shares in a matter of a few clicks on your trading screen. Similarly, you can sell your investments online. You can transfer your money electronically back to your bank account.
Few international brokerage firms like Interactive Brokers, TD Ameritrade, Charles Schwab International Account, etc. permits Indian citizens to set up an account and trade in US stocks, mutual funds, etc. 
While investing in international stocks, you have to make transactions in foreign currencies. For example, if you are trading in the US stock market, you have to pay the brokerages in the US dollar. 
As per the RBI notification in the Liberalised Remittance Scheme (LRS), an Indian resident individual can only invest up to $250,000 overseas per year. 
An Indian investor has to pay 20.60% of tax on gains of foreign equity. There are no short or long term tax gains from foreign equity gains. If you open an account with a foreign broker as mentioned above, you are liable for taxes in both countries subject to the right set-off per Indian tax laws.