SWTOR doesn't really have a discrete Tutorial portion (e.g., "how to move," etc.) that a player might skip on subsequent characters. Across the various classes, however, the early levels cover a lot of the same ground, except for their storylines.
The storylines are central to the game. Skipping levels on your 2d, 3d, etc. characters doesn't work because you'd miss out on the early storyline scenes. But there should be some way to expedite the somewhat redundant parts of leveling on the starter planets (Hutta, Korriban, Ord Mantel, Tython). Here are some possible methods that be unlocked for later characters.
They might even be limited according to the first character's story elements, instead of any story completion. Possible triggers could include (a) finishing any character's story line; (b) achieving a certain Legacy lvl; © finishing a storyline for a character from same faction (Empire/Republic); or (d) finishing a character storyline that began on a particular starter planet. Whatever the predicate, the idea is to streamline some early progression but not actually skip that content.
Here are some options for these later characters:
1. Sprint is available at lvl 1. There are good reasons to do this for all characters since travel is a bit of a time sink in the game. Even if SWTOR increases the overall base movement, allowing second characters to move relatively faster will speed up the early levels without significantly changing the quests, layout or content. You'll still do all the things on Hutta, just a little faster.
2. Start with full rest. Again, the second character won't avoid early content, but kicking off with max rest would help speed them along with a benefit that burns off by the time you get to higher levels.
3. Global Cantinas. Another indirect way to spur a second character's progress is to allow them to gain rest throughout the starter planet. So if all second characters on Korriban gain rest irrespective of where they are, it could push their progress forward. This might only work in conjunction with a starting rest bonus (pushing the leveling advantage past the starter planet), since you'd have to start the character and park it on the planet for awhile to build the rest bonus. You can already do that in the starter planet's Cantina, so this alone is not much of a gain. If you started with full rest and kept that status until you left the first planet, that would be a pretty good bump. It also would wear off once you got to the mid-teens, which feels about right for this kind of shortcut.
Food for thought