I would imagine that most of the big ships, don't travel around much, except for displaying the Imperial flag sort of public relations bit, much like most battleships or ship's of the line, from 1800 on. These ships would pretty much stay in port (orbit in a specific system), with constant make-work, battle drills and of course various competitions like small boat crews, boxing matches, gunnery competitions, etc.. After all, if you are sending a communication to a fleet, you have to know where to send it and since the Imperium is much like the 1800's navy's with captains having a broad power because of lack of communications from home, so you keep them where you can find them. Of course the enemy also knows where you are but then the same is true for them as well, you pretty much know where their big ships are.
Now the little ships, cruisers and destroyers, they are often moving around, within their assigned area of patrol. The Navy would probably jump a small destroyer or cruiser squadron into each solar system in the sector for a irregular stay, if they even let the system know they are there. A good way to find pirates, illegal trade or the like. I would probably increase a fleet's size to double or triple the number of smaller ships that it would normally have.
I would also have a fleet arm that would move around a lot, i.e. yearly. They would jump into an area, often in a minimal system, watching the trade and being where they could be easily summoned in case of need, often by having couriers in the surrounding systems that they could reach in 1-2 jump-4.
Of course, the closer to Home World you got, the older, bigger the ships would be and the admirals and such a lot more political, as much as you allow in your game.