Ship speed at 5/5 - no more ?

Yes, you can make ships move faster by strapping on extra engines on a ship, progress into the engine technologies, or making your units move with a ship carrying a Fleet Warp Bubble projector.

Note: ships in a fleet will move at the rate of the fleet's slowest ship.

You've been hit by a Mega Event that restricts all ship speeds to a maximum of 5 for the duration of the game.  There aren't a lot of things you can do about it, but it only takes effect on the turn after a ship is built, so your first turn with each ship will be full moves.  You may want to consider using all-engine designs in order to move your ships the most distance possible in that turn, and then upgrade them into what you really want them to be-although this will be more expensive.

In general, you probably want to turn Mega Events off, as this isn't really avoidable otherwise.

While this particular event is obnoxious, I wouldn't say that it in itself is worth disabling Mega Events for, if you like other ones. I've seen it maybe three times in 90 or more games, so it's not incredibly common.

And on a big map I'd reload like a shot the instant I saw it.  Move of 5 -- sheesh.  

Although here is a tip that no one brought up yet:

You can cheat the system by having ships with fleet warp bubbles in your fleets, by doing so, your ships can move up to 8 squares per turn, instead of the usual 5.

Don't ask why, but it works.