Hi Silva,
20 years isn't old for a piano. If your piano is going 'bong' rather than 'dum' then it is probably much older than 20 years, or otherwise has been kept in very bad conditions (damp, etc) at some stage. I take it that your strings are dull and corroded, rather than bright and shiny. Just for your own curiosity, you could ask your piano tuner whether he thought re-string would be worthwhile and what the approximate cost might be. It might be possible to greatly enhance the action with some skilful regulation, too, giving you something closer to the feel of a new piano. But, my view would be that unless it was once an expensive piano you are probably better off replacing your instrument.
Therefore, go to your nearest piano shops and try some instruments out. Check the price tags to see what your money will get you. Bear in mind that good discounts off the manufacturer's list price are usually available, for instance I know of one UK dealer who advertises 30% of the list price of all Yamahas as a matter of course. Therefore, check what the manufacturer's price is before you make a purchase, and shop around.
To get your search started right now without having to leave home, a quick search of the internet should reveal several UK dealers' websites, some of which give pricing information. If you find that new pianos take you over your budget, then consider a used instrument. In some instances you can end up with better quality for less money. Whatever you decide to do, let your playing experiences of prospective instruments be your guide.
Finally, get some lessons - a good teacher will make a huge difference to your progress and musical understanding, imho.
Happy hunting.
Richard.