A main reason to avoid mixing them is travel speed. Also, if you have some alien ships, keep in mind that having a fleet of mixed fuel (caslon and hydrogen) can be a pain. To see what kind of fuel a ship needs, click the design (in the selection box) and look towards the top right part.
The single issue with using big ships I can think of is that they struggle with chasing down ships. But it is incredible what a few carriers and some jump inhibition devices can achieve.
In my current game I design my own ships, but nothing outlandish. In my previous I left research and design to the AI. My method is fairly similar.
1: I micro manage fleets. This is a choice. I set them all to defense and "current system only". I don't want them to take off on their own. Look at the strategy (war?) forum for a guide on automizing this.
2: Destroyer fleets are great for mopping pirates. Often 4 ship fleets, but in my current game I am (was) poor, and used two destroyer fleets with maybe a bit stronger destroyers than the AI designs (300 size, no troop compartments). They manage pirate bases just fine in early game (heavy rail guns and basic but upgrades torpedos). This is Zenox with a nasty shield, though, usually I have gone with 4 destroyers for this. They will lose attacking a "rich" pirate gang of some success if all ships are at home.
3: The silly Sluken declared war. Beyond panic building troop transports (in their own fleets if in a fleet) to reinforce their target, I built some cruisers (400 size which is max). I think I built 10 and split them in two fleets, one with 6 cruisers, another with two destroyers and an alien cruiser to make 7 ships. If I was not so poor (very hard, corruption, and harsh home world), I would probably build a couple of two 8 cruiser fleets at this point. Add a couple of carriers if they are available (which they are for me now, but I have not bothered to build).
I usually label my destroyer fleet "Raider XXX", a main fleet "Home fleet", and cruiser fleets "Battle xxx". I number the battles 2, 3, 4, may give them a further name to identify them (east, west), and bind them to the key matching their number (ctrl 2 to bind to 2). Home fleet is 1. Raider fleets are good for munching bases and protect against pirates, Battle fleets do the main work.
Now, the silly Sluken has lost their 3 colonies in my direction (we are neighbors in a ring), and the capital is next. I don't want that, I want the war to end. I wonder how to do this, and could probably build another 15-20 cruiser, bust their capital space port and blockade it. That usually makes the AI consider a peace for free
The toy I got that the AI designer will never build, is a 300 size explorer with a long range scanner. Currently parked in an empty system a bit on my side of the Sluken capital, reporting on fleet movements and enemy mines. I kill the mines I can find, and the scanner shows fleets in mysterious directions, pointing at other fuel stops. Such ships are also nice for scanning areas for pirate bases.
Of course, battle fleets of 8 cruisers are quite early game of limited ship sizes. They typically evolve to 8 cruisers and 2 capitals, 6 cruisers and 4 capitals, 6 cruisers and 8 capitals (with always 2-6 carriers in addition if available). After a while I will probably have only capitals, carriers and destroyers (and other special ships like troop transports, resupply ships, and scanning explorers).
At some time in past games I have become filthy rich, bored, and I wait for something nasties to arrive (which they never do). This is the time period where I order 120 automated frigates to take care of trivial matters. And 100 more if that was not enough.