Can you access the entire ship via the Turbolift that services the bridge? Is it a single system, or do you have to switch lifts somewhere along your journey?
The turbolift system on the USS Enterprise-1701-D went through the entire ship longitudinally and vertically. The entire system supported a maximum of twenty cab vehicles simultaneously and required a turbolift control system which could be managed from the bridge.
In a Galaxy-class starship, turbolifts are regulated by turbolift control and an average of ten turbolift cars are in service at any one time. Some peak usage times, such as change-of-shift, can call for a doubling of the turbolift cars with only a 22% decrease in efficiency. The reason this is possible is because the turboshaft network is designed with the specific purpose of allowing multi-access loops and thus permitting a flexible route for each turbolift car. -- Memory Alpha -> Turbolift
Section 13.6 Turbolift Personnel Transport System - The turboshaft network consists of two parallel main vertical arteries, which provide service to hubs in horizontal networks on each dec. Redundant horizontal links on Decks 8, 10, 25 and 31 connect the discontinuous vertical segments. The network is designed to provide alternate access routes to all decks, permitting alternate routing during times of heavy system usage. The design philosophy also minimizes the effect of any given single malfunction on overall system performance. Additionally, there is a single dedicated emergency turboshaft connecting the Main Bridge on Deck 1 to the battle bridge on Deck 8.
Memory Alpha has buried in its images something that appears to hold the answer, as I don't recall if it was ever made explicit during the run of the show.
These are schematics of the secret Prometheus-class starship, which first appeared in VOY 4x14, Message in a Bottle:
You can pretty clearly see the turbolifts in it are all connected, so you can get almost anywhere on the ship with one ride and a little walking.
These are similar schematics for the Galaxy-class Enterprise-D:
It's much harder to make out, but there are similar passageways that turn at right angles, that go throughout most of the ship. It appears as though there are 2 main ones, one exiting the bridge and going straight down through the decks of the saucer section, while the other goes down a few decks then veers off in the direction of Engineering (then splits into 2 paths in the drive section).
Now, using that first image of the Prometheus-class starship as a baseline, you can see the turbolifts themselves as white/grey blobs in the passages. Here's two close-up shots of the Enterprise-D:
There's numerous similar blobs in several passages, except these passages aren't connected to each other. This would seem to imply to me that there are several disconnected turbolift systems on the Enterprise-D, even though it's not a limitation of the technology.
More likely, it helps with efficiency. One set of turbolifts would go around the circumference of the saucer section, another would express between the Bridge and Engineering, another would act as an elevator/lift... And it would help keep the children from interfering with the Starfleet personnel who are on duty.
Note: Those parts of the last two images that are highlighted in red and purple are the plasma and replicator systems. There were issues with them in the episodes those shots were taken from. It is unrelated to this question/answer
The turbolifts go both Vertically and Horizontally thru the ship.
If you read some of the history in the linked article, you will see that they were originally likened to Cabs, and there were designs that showed a map of the ship to make it clear that their destinations were more than just points on a single vertical shaft.
The TNG episode Contagion showed Geordi being tossed around as the Turbo lift changed directions repeatedly. (Not just up and down, but side to side as well.) You may notice the moving lights give an idea as to what direction it is going; TOS did something similar with both lights and sound changing as the turbolift changed directions.