(This list is not complete. It just contains a few of the best tools to get you going.
All software is freeware)

Display subtitles:

Simply use a player that can display subtitles out of the box:

Video Lan Client (VLC)  (Windows, Linus, MacOS)

GomPlayer (Windows)

MPlayer (Windows, Linux, MacOS and more)

or use your favorite player together with a subtitle add-on:

DirectVobSub/VSFilter (Windows)

Perian (together with Quicktime Player) (MacOS)

Rip subtitles from DVD or SUB/IDX files

subrip (Windows) (can even rip hardcoded subtitles that are part of the video)

Subresync, part of the VobSub Package 

D-Subtiler (MacOS)

OGMRip (Linux)

Edit subtitles

Gaupol (Windows, Linux)

Subtitle Workshop (Windows)

I recommend Version 2.51 but there is also a newer beta version that is totally different. (I like the older version much better!)

Jubler (Windows, Linux, MacOS)

Submagic (Windows)

Other useful software for Windows:

VirtualDub (Video reencoding, editing, joining...)

AVIAddXSubs (Merges multiple .srt or IDX subtitles with avi that will play on newer DIVX DVD players. This is a great way to get Arabic, Hebrew or Chinese subtitles without reencoding the video.

 Wanna learn more?

 Browse through our Tutorials and have a look at the Subtitle Help corner as well. Your question might have been already asked and answered.

 Still none the wiser?

Head over to videohelp.com to find more amazing and free software.
They also have a huge forum. If they don't know it or can't figure it, nobody does.

And now:

Welcome to the world of subtitles.
Now you are ready to find out how deep the rabbit hole goes.