The Original - Some stories hold up much better than others. The good episodes watch like a classic film where the stories are good enough to ignore differences in style and production value. Most episodes show their seams a bit more but are still entertaining. Some are either horribly dated or just plain campy. The Original is probably the least consistent of all the series, both in terms of continuity and quality.
The Next Generation - 100 years after the Original Series with fancier tech but same idea of exploration. Seasons One and Two feel like the Original but with worse characters and better production values. Season Three the show starts to find its voice. TNG is consistently watchable and entertaining if not excelent starting Season Three with characters who developed just deep enough to make interesting stories, although they can sometimes feel a bit two-dimensional. Many of the staff writers are very good and craft great stories within the confines of the setting.
Deep Space 9 - Set on a strategic outpost in the star system of an important species that wants to join the Federation, near a major enemy to the Fed, and also near a fast passage to a location halfway across the galaxy with a new enemy for the Federation. Deep Space Nine builds on the universe The Next Generation laid down and adds characters that are deeper and come with more inherent conflict. The show also has a very large cast of recurring characters that may play a very large role in a few episodes, but are not part of the main cast. Seemingly one-time guest characters often come back. It also has a story that builds from beggining to end, making the series feel tighter than others. The First Season has a few great episodes but most of the First Season is much weaker than the rest. Don't listen to the critisms that the show is bottled on the station and doesn't go anywhere; three large warp-capable shuttles and later a warship permantly assigned to DS9 allow the show to go to many, many other locations.
Voyager - Set on a ship that got sent to the other edge of the galaxy, kind of a Lost in Space but Trek. The pilot sets the show up for inherent conflict and interesting dilemas such as worrying about supllies like nuBSG. Instead, starting in the second episode, you can see the show slowly zip down its fly and piss on all this. The show has very good production values in terms of sets and special effects. The characters change personalities like hairstyles and the writers could not even remember basic facts about the characters like their age (Tuvok's age is given three times, each time different, and not to account for aging). The show is consitently slickly produced and postively vapid with a few good episodes that are the exception.
Enterprise - Set 100 years before the Original Series on the first Earth starship fast enough to go exploring. Season One and Two feel like Voyager on a new ship with characters that are even flatter, if such a thing were possible (Though to be fair most of VOY's characters weren't as flat as their personality was completely malleable to suit the writer's whim of the week.) During these two Seasons thw whole prequel thing is totally wasted. Season Three experiments with DS9-style continuing plot arc and a little more militarism, which has mixed sucess but does a lot to improve the show. Season Four actually does its work as a prequel and has interesting stories by virtue of great guest characters.