Travel between star systems was a routine fact of life for most of the cultures Blake and his followers encountered. The vast majority of starships employed some variety of FTL (faster-than-light) drive, for example all the ships of the Federation, System, and independent human travellers.
It was the same for most alien civilisations, with a couple of exceptions. The capsule encountered by the Liberator en route to Saurian Major appeared designed to travel interstellar distances with the crew in cryogenic suspension. The Keezarn people had launched a ship, apparently at STL velocities, which would carry a teleport terminal to a new planet for them to colonise.
How ships achieved FTL velocities was not revealed. While the London was fitted with a 'hyperdrive' and there were occasional references to hyperspace (Ro mentioned hyperspace ships while Blake suggested the great speed of the Liberator might be a result of accessing negative hyperspace), ships were seen travelling in normal space - and the terminology involved suggests a degree of time dilation (experienced in normal space while experiencing a high degree of STL acceleration) being experienced. However Dayna only stated that it was impossible to achieve extremely high speeds (Time Distort 12, in this instance) in 'real time', suggesting that the actual distortion involved in use of most levels of Time Distort speed was negligible.
At standard cruising speed it took eight months for the London to travel from Earth to Cygnus Alpha. No other journeys appear to have taken anything like such a long time, at least from the viewpoint of the crews involved, with travel times most commonly measured in hours or days. Time seemed to pass at the same rate on planets and ships, further lessening the likelihood of time dilation being a factor.
It seems that travel between galaxies was not yet possible using existing Federation technology. On discovering the minefield beyond Star One on the galactic rim, Avon speculated that an intergalactic drive had been developed. The Andromedans almost certainly possessed such a drive.
Time Distort was the standard measure of interstellar speed employed by Federation ships. The exact speeds involved are difficult to determine exactly.
The Mark Two photonic stardrive could achieve Time Distort 15, a considerably greater speed than had previously been thought possible.
Zen measured speed in terms of the 'Standard' convention - i.e, Standard by One, Standard by Two, and so on. If 'Standard' refers to the standard cruising speed of the Liberator then, rather bemusingly, the crew routinely travelled around at five or six times the ship's 'normal' velocity (this may be why the energy banks seemed prone to exhaustion on occasion).
On the other hand, 'Standard' may refer to some other constant, but given the great speed of the Liberator this is almost certainly not the speed of light.
The maximum speed the Liberator could achieve unaided was Standard by Twelve, although the second DSV-class ship launched by the System achieved Standard by Fourteen immediately prior to its destruction. Blake once said that the crew would lose control of the ship at Standard by Twenty, suggesting greater speeds were achievable in the correct circumstances.
Even after transferring to the Scorpio, the crew continued to use the 'standard' convention - which means the measure would have to be modified given the much lower standard speed of the Scorpio at that time, or (more probably) it did refer to an external constant.
Converting between Time Distort and Standard speeds can be problematical: early instances of the same vessel's speed being given a Time Distort and a Standard value in close succession suggested Standard by One was effectively between one-and-a-half and two times Time Distort 1.
However, the Mark One photonic stardrive developed by Plaxton was calculated as having a top speed of over Standard by Twelve. Plaxton and Atlan later referred to it having a top speed of Time Distort 12, suggesting that at this time the two systems were interchangable. (Possibly 'Standard' speed did vary from ship to ship.)