How to actually invest in an index fund?

Usually, I go with a broker (share trader) that deals specifically with buying and selling securities. Looking at this page gives a list of possible brokers in the EU and elsewhere.

The other way to go is to choose your index fund and invest directly with the company that operates it. For example, I believe Vanguard offers this in the UK.

Opening an account with either type usually involves opening two accounts: a brokerage account (to hold your investments) and a bank account (to hold your cash). Once you fund your cash account with enough money, you will then be able to purchase any security (including index funds) that you have enough money for.

Be aware of things like: inactivity fees, brokerage fees, and the costs of owning the index fund itself. They all eat into your gains.

In Germany many brokers offer savings plans for free, I guess this applies also in other countries. When you want to invest small sums, you can buy your portfolio with one or more monthly savings plans which invest a fixed high amount each month, say 500€.

Smaller savings plans are IMHO a good way to start with the stock market, as an alternative if you don't want to invest your savings at once.

If you look to invest a higher amount, compare what the brokers ask for - check for free buys or low fixed fees.

Around here (in Denmark) most banks offer a funds trading module in their online banking solution. You put in an order to buy or sell so-and-so much of such-and-such asset, and they carry out the trade and settle it over your current account.

There are fees, yes. My bank charges 30 DKK (about $5) for each trade (or 0.15% of the value, whichever is more). But that's just transaction fees; holding securities does not cost me anything.

This would make it a non-starter if you're doing speculative trades, certainly -- but if you're just looking to buy-and-hold some mutual funds, the convenience relative to having an actual brokerage account somewhere else may well be worth those fees to you.