There's nothing wrong with Panther for that machine--I haven't heard much about the speed of Tiger on G3s. And if you do get Tiger, you'll either need to find a DVD drive to install from (unless your iBook has one built-in), or go through the hassle to get the CD set.
In any case, as said, you NEED more RAM. I'm surprised you've even got 10.2.8 running on 128MB--it must be painfully slow. I've found 160MB to be the bare minimum on 10.2, and I wouldn't even try 10.3 with under 256MB--it won't start to feel right till you give it at least 384MB.
You should be able to add 256MB for about $50, and 512MB for around $100, and that's probably going to make a bigger difference than the OS upgrade in terms of speed.
Once you've added some RAM, installing 10.3 should be very easy: Back up your data (to be safe), buy a copy (can be had quite cheap now), pop in the CD, follow the instructions. Might be worth doing an "Archive and Install", though I ran an "Upgrade" install for the full life of 10.3 without any major issues.
10.3 won't likely get you any more speed, but if you go straight to 10.3.9 you'll have a nice version of Safari and a pretty stable system.
