You've got three major variations of this campaign archetype.
- The Tramp Freighter: The ship is a working commercial vessel of some sort (usually a small-scale cargo or passenger vessel, working as a for-hire charter vessel or along adjunct lines tied to major commerce lanes). The scope of the campaign is often "local" (in space-faring terms; working within a solar system is the lowest end, within a small stellar region or zone the typical end, w/ greater scope being very rare). Common gameplay activities revolve around keeping the vessel operational, keeping themselves in good health, and keeping their difficulties away. Players assume the roles of characters with little or no augmentation, supernatural power, or out-of-the-ordinary technologies. (So, we're looking as the sort of table where being a City Rat is not out of line, and being a Vagabond is a viable option; ordinary military personnel in their ordinary capacities are a significant challenge, and better outwitted than outfought.)
- The Exploration Expedition: The ship is a public or private vessel configured for (if not purpose-built for) long-term, and often long-range, exploration. The scope of the campaign is often very far-ranging, and both the crew and the vessel will be trained and equipped to operate in a self-sufficient manner. First Contact is a common gameplay activity, and with it the potential to engaging in a wide variety of diplomatic and economic negotiations (and their intrigues) with heretofore unknown parties. While the nigh-platonic form is classic Star Trek and it's Five-Year Mission, taking inspiration from real-life histories of exploration expedition is a good idea. Players assume the role of key officers, and often take up alts of lower status who are more likely to engage in shore expeditions or otherwise do stuff that matters away from the ship. As with the Tramp Freighter, the characters are unlikely to be superhuman or super-powered (in relative terms, if not absolute ones) and the same will be--for the most part--true of the technology at their disposal at any time.
- The Warship Campaign: The ship is a warship, engaged in warfare operations. In this respect it's a shipborne variant of the Mercenary or Military Campaign, but as with the others the ship as both homebase and focal point is part of the paradigm. Campaigns of this sort take their cues from Space Battleship Yamato, various Age of Sail series, and the real-life accounts of ships such as the Enterprise in World War II. Their activities focus around military operations, so there is an overall objectives to keep in sight, and there is a determined opposition out to stop them by means of blowing them out of the sea of stars. Combat--fleet and personal--is a common occurrence, but this need not be a series of fights; real warfare is about objectives, not slugfests, so it's about getting yours before the other side gets theirs. Players also assume the roles of key officers, and should take up alts that do stuff the key officers wouldn't do.
- The Ship: Whatever form it takes, the ship is a character unto itself and should be treated like one. Even if the stats for it enter play as any other of its class, that specific vessel is specific to that captain and crew and should be individuated accordingly. ("There are many like it, but this one is ours.") Players should value the ship as the expensive, mission-critical thing that it is and not be cavalier about its welfare. The ship is homebase for the characters. It's the focus of their operations. It's the center of their lives, without which they're unable to carry on at all.
- The Travelling: It's not a starship campaign if you're not sailing the sea of stars. It's a naval game, so get out of port and out in space. You should be visiting new ports of call on a regular basis, with only a few locations being seen frequently; regular visits to a known friendly port is something that depends on the specific set up of your starfaring campaign (tramp freighters and exploration vessels will visit on very different rates of regularity and frequency). Being that this is RIFTS, it's going to be more Space Opera than Hard SF, so you've got that going for you; don't be shy about the planets and what's there.
- The Other Starfarers: Your opposition is very likely to be as spaceborne as your allies are. This means that you and your players need to figure out how your table wants to handle ship-to-ship interactions (combat and otherwise) to ensure that the players aren't bored when time comes for the crew to make the ship happen. This will differ depending upon the ship; tramp freighters with some guns attached are not the same as a naval dreadnaught in command of a squadron or even a massive fleet. (Yes, you can get your Legend of the Galactic Heroes on here, so go for it.)