I am an amateur player and from time to time I like to play chess. Is it good to start playing against a computer and try to find an Elo setting where the amount of wins and losses I score is roughly the same? If so, which chess engine is the best for that kind of thing (especially for linux)?
State of the art chess engines will win every game against an amateur, so they won't tell much about your playing level. On the other hand, the engines of sensible strength (or a handicapped engine or similar) I've tried do not emulate human playing at amateur levels well. While there are stronger and weaker engines and most probably you will find an engine against which you win around 50 % of your games, I wouldn't say that tells much about your level, because your playing style may affect the results a lot. For example, I've found that it is often easy to beat most amateur-level engines just by exchanging most pieces (while making sure you don't fall into any tactical traps) and winning the endgame the engine won't understand anything about. If you practice too much against a single computer opponent, you might develop bad habits.
If you're not willing to join a chess club, you can play chess games against real people on one of the various Internet chess sites that have a rating system. When you get a reliable rating there, you can try to see (e.g. searching in the forums or help files of the site you chose) if there is an approximative formula that gives you an estimate of what your Fide rating would be (that's what you asked for).
Please also note that the rating scales and systems used by different Internet sites, by your national chess club, and by Fide may differ much. Even if one site calls their rating "elo", it means only that they use the Elo rating system for the ratings. However, even the rating pool there has no connection to the Fide rating pool, and the ratings do not mean the same.
A beginner player has no ELO rating. I think computers are terrible opponents unless you're willing to take a drubbing every game. Download the free Stockfish and learn to use it. You won't enjoy the experience. I am a slightly above-average club player. I have no chance against someone who has no chance against someone who has a sliver of a chance of defeating this engine. I use Stockfish as an analysis tool.
I recommend you join your local chess club. If you're in the USA, go to this page and find a club. There you will find other opponents of your skill level, mentors, and comradery. You can play in tournaments and earn a USCF rating, which is similar to ELO.
There are lots of websites on internet which helps you to check your ELO rating like http://www.chessmaniac.com/ELORating/ELO_Chess_Rating.shtml
You can also try http://www.forwardchess.com/ to improve your chess skills which will definitely help you to improve your ELO rating.
The http://chesstempo.com tactics server provides an estimate of FIDE rating based on your tactical performance.