I want someone to recommend me fonts that are "in" now-a-days and should be available on most platforms. You see, I am a web-programmer and not much of a web designer. I am asked to use serif fonts.
I have found 8 Definitive Web Font Stacks by Michael Tuck a good read to start with.
I'd generally steer clear of serif fonts as I don't think they work too well for web content. Having said that, Georgia isn't too bad as a choice if it's got to be serif.
I'd second the recommendation for the SitePoint font stacks article already mentioned, and also point out that typography is more than just picking the right fonts - there are a couple of excellent articles recently published that are well worth a read:
It's also worth mentioning that css3's @font-face is very usable these days with surprisingly good browser support (See CSS At Ten and Typekit for more info, and the excellent modernizr javascript library to help with cross browser (that is IE) support).
Try pairing serifs with sans-serifs. For a classic pairing consider georgia for headings and verdana for body text.
You might also want to check out Type Kit if you would like to use non-standard fonts without having to use Fahrner image replacement to display non-standard fonts.