Warning: Be prepared for spoilers. This writeup is chock-full of them.
Virtually everything changed in Series 3. The theme changed from Space Odyssey-esque music to rock guitar. Kryten returned as a regular, and Holly's face was changed. The sets were changed inside Red Dwarf - the Drive Room was now the Science Room, Rimmer and Lister's quarters were now labelled "Officer's Quarters", and the whole interior had been given an overhaul (most of it was now a beige colour).
The costumes were also changed. Lister went from slobby RD uniform to leather jacket... but he kept that stupid hat. Rimmer went from neat, clean RD uniform to bottle-green. He even had a hat with a microphone attached, but that was lost after this series. Cat, of course, changed his suit all the time, and Kryten hardly ever changed.
On to the episodes. "Backwards" was similar to the book of the same name by Rob Grant in that the characters made their way to a backwards universe. However, the teenage sequence in the book was only touched upon in the show. Also included was the 'Wilma Flintstone' dialogue that had been cut from last series' Queeg.
"Marooned" crashed Lister and Rimmer onto a snow planet without them being able to take off, and with little food, water or fire fuel. This was a low-budget episode, and was shot first. Look out for Kryten's first line (Blue Midget is loaded) - producers tested Kryten out on many accents, and decided on Canadian... though he seems to go through them all with this line.
"Polymorph" was the first Red Dwarf episode rated M in Australia, and probably something similar in the UK. A shape-shifting mutant climbed aboard Red Dwarf and sucked emotions out of the crew - even Rimmer, who was supposed to be a hologram. To this date, I do not know why it is rated M - for scenes with the polymorph, or the boxer shorts scene.
In "Bodyswap", Rimmer convinced Lister to swap bodies - just so that Rimmer could pork out on his favourite foods. It was the first episode shot without a studio audience, because of all the redubbing of voices by actors. Thankfully never shown was a painful sequence for Robert Llewellyn - Kryten was to produce a flame from his fingertip, but the device repeatedly electrocuted the actor. The scene was cut, and lost in the archives.
Every series seemed to have an episode which mucked around with timetravel and causality. "Timeslides" was the one from Series 3. Kryten developed photos and slides in a mutated mix that made the pictures spring to life... and gave people the ability to walk into the past. Naturally, causality was screwed around with a couple of times... even to the extent of Lister being Adolf Hitler's near-assassin!
"The Last Day" showed Kryten's possible departure... again. He was to be replaced by a robot called Hudzen, so the Dwarfers had to band together to get rid of him. It was a bit of a relief for Robert Llewellyn, as he got out of the Kryten suit for a while and into a much more comfortable tuxedo.
So that was series 3. Click here for Series 4... or here if you missed something from 2.