I noticed that rpm takes long time to settle at idle. It stays at 1500 rpm for few seconds before finally coming down to about 800-900. Also I noticed that my timing belt has way more free play then it should but the belt itself doesn't look worn or damaged just way too loose. Is replacing timing belt myself recommended and if yes how hard or difficult is it to do it or should I seek mechanic help ...
Hyundai Tiburon 2001 138,000 miles

Well the idle is perfectly normal to be up a little bit before it settles down.

As for the timing belt, it should be replaced every 60,000 miles, so it should have been done twice on yours already. If it wasn't done I'd definitely get it taken care of. I've never messed with one myself, and though it doesn't seem to be incredibly difficult I'm not sure I'd want to risk messing up the whole engine by doing something wrong.

On a side note, your Tiburon is not a GK (2003-2008) so this really isn't in the right place.

l think you may be getting confused between the timing belt and the normal drive belts. the timing belt runs under a plastic cover at the right hand end of the engine and you can not see it until the covers have been removed, as the above post states your belt should have been replaced twice over by now. l think you should get a reputable garage to check it over for you to be on the safe side

l am also moving this post to the correct forum

Yes, change the timing belt immediately. You are lucky. If the timing belt breaks, your engine will be destroyed (bends the valves)....very expensive to fix. Also, your gas mileage should take a good leap, if your MPG has been low. Don't do the job yourself. If you set your belt wrong, the timing will not be right....& you'll destroy the engine again. Yes, your MPG can return as if the engine was new.....if your engine is still good.