If you want partial automation but to retain some control, look for a guide to Illustrator or Photoshop actions (ideally, Illustrator working with logos in vector format rather than Photoshop, so it's possible to have consistent variants at other sizes, for print, etc etc). There are loads of guides and tutorials out there. The basic principle is, actions are a bit like macros. You record what you do, then you can do it again with a button press or keyboard shortcut.
If you want to program complete automation for raster images headed for the web, the tool would be ImageMagick. Open source, requires some programming skills to use.
If you want to make good logos, then PLEASE PLEASE PLEASE DON'T automate any part of logo creation or formatting unless there are very very unusual rare circumstances (maybe if you need to put something like a white outline around lots of existing logos before putting them on some logo gallery site... can't think of any other example).
If you're creating logos, it's seriously not a good idea to blindly trust automation at any stage instead of spending mere minutes getting that effect right for that case. A logo for even a tiny company will be seen tens of thousands of times, hundreds of thousands of times a year. Even divided between each variant, that's a lot of views, so small investments of time will pay off.
Also, if the variants aren't sufficiently consistent that they are instantly recognizable as different forms of the same thing, it can (if you are lucky) seem sloppy and make a company seem less professional or (if you are unlucky) confuse people who don't realize this is the same company they encountered before, losing the trust-building familiarity that is one of the main benefits of having a good logo. Varying logos should be done with care (see for example this question on logo colour variants).
9 times out of 10 effects like drop shadows in a logo will be a bad idea that would add noise and reduce clarity (see this question on effects in logos for more detail). Assuming yours is a rare case where it does make sense, at the very least, consider using something like actions where you can check and tweak that each time it works. Even if you're just batching a load of other people's logos - you'll look bad if someone else's logo is noticeably worsened on your site.
Logo design is something where it really, really pays to get it right first time.