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If you get caught without a valid ticket, the official fine on the spot is HUF 6000, which is about 20 €. Do not let yourself be fooled with another amount, it is HUF 6000, and the received paper is valid for traveling that day. But if you are cool enough you can get away without paying at all.
At around 20:30 or later, the ticket inspectors that stand at the entrance to the escalators go home, meaning you can get on the metro for free. Be aware also that if they have controls while you exit the station, they announce it as you get off the train and again every 2 minutes or so ("Please have your tickets ready for inspection as you leave the station.") If this happens, simply get back on the next train and get off a stop before or after your chosen stop. Do not try and run past any inspectors if they have a control - they are notoriously aggressive and will have no problem chasing you.
On the trams and buses the inspectors legally aren't allowed to touch you. Avoid speaking English or German to the controllers. Instead, pick the most obscure language you know and keep them busy until the next stop. When the doors open, simply exit the tram and walk away. Never give the controllers your ID or passport. Maybe try searching for it until the next stop, then get off the tram/bus then.
Some of the controllers will try to take you to a cash machine if you say you have no money. Same system applies - simply walk away from the controller. I've also heard of people stalling the controller for so long that the controller gets bored and walks away.
On the green line there are security guards stationed on the platforms, but don't worry, they won't check your tickets. They're just a security precaution. The same thing applies on tram 4 or 6 at night (they run 24 hours). They will stand at some entrances to the tram and check peoples' tickets as they enter the tram (They'll be wearing either a yellow vest or a green armband.) Choose a different entrance and you'll be fine, because the controllers stand at one door and don't move through the tram.
For finding connections between two points in the city, there's a very good route planner available also in English: http://www.bkk.hu/en/timetables/[http://www.bkk.hu/en/timetables/]