I don't want to get involved in trading chasing immediate profit
That is the best part. There is an answer in the other question, where a guy only invested in small amounts and had a big sum by the time he retired. There is good logic in the answer. If you put in lump sum in a single stroke you will get at a single price. But if you distribute it over a time, you will get opportunities to buy at favorable prices, because that is an inherent behavior of stocks. They inherently go up and down, don't remain stable.
Stock markets are for everybody rich or poor as long as you have money, doesn't matter in millions or hundreds, to invest and you select stocks with proper research and with a long term view. Investment should always start in small amounts before you graduate to investing in bigger amounts. Gives you ample time to learn.
Where do I go to do this ? To a bank ? To the company, most probably a brokerage firm.
Any place to your liking. Check how much they charge for brokerage, annual charges and what all services they provide. Compare them online on what services you require, not what they provide ? Ask friends and colleagues and get their opinions. It is better to get firsthand knowledge about the products.
Can the company I'm investing to be abroad?
At the moment stay away from it, unless you are sure about it because you are starting. Can try buying ADRs, like in US. This is an option in UK. But they come with inherent risk. How much do you know about the country where the company does its business ?
Will I be subject to some fees I must care about after I buy a stock?
Yes, capital gains tax will be levied and stamp duties and all.