Greetings CMDR,
I am not an elite player, but I have a few hundred thousand ly behind me with my fleet carrier around the galaxy.
I'm one of those CMDR fools who instead of flying light and fast around the galaxy and making lots of credits, prefers to move slower with his fleet carrier spending and throwing out the window millions of credits earned from the same exploration, everyone has his own taste!
My advice is to eliminate all services that you do not need on your fleet carrier and fill it completely with tritium.
It doesn't matter if part of that tritium you will consume to transport the same tritium around the galaxy, you fill it completely, without leaving a single space free!
Just before leaving, take a ship with lots of cargo (I have an imperial cutter with 792t of cargo).
Fill your cargo ship with more tritium and land on your fleet carrier.
792T of tritium in your ship's cargo will allow you to make the first 6 jumps to get at least 3000ly away from the bubble, a distance in my opinion minimum to be able to start exploring in earnest, without depleting the precious reserve of tritium on your fleet carrier.
As for mining tritium, as repeated many times by CMDRs more experienced than myself :
1) The best method for mining tritium is not to mine tritium!

2) If you are out and about in the galaxy with your fleetcarrier, I recommend going through Colonia to buy some overpriced tritium from other fleetcarriers so you make some money from other CMDRs and the economy grows!
3) If, on the other hand, you find yourself lost in the galaxy with your fleetcarrier with a few thousand units of tritium and tens of thousands of ly away from the nearest station, and you find yourself in the only situation where it makes sense to mine tritium, here are some tips :
Look for the usual icy rings in a system you have just discovered, big chance that there is a tritium hotspot, go to the hotspot and what you are looking for is a slow spinning asteroid, very slow, almost stationary, this asteroid must have 3 or 4 Tritium Subsurface Deposits, don't stray too far from the centre of the hotspot and don't spend too much time looking for an asteroid with these characteristics, if you DON'T find it in a short time, leave and find another hotspot, the galaxy is full of them! If you find an asteroid with these characteristics, you can start launching your Sub-surface Displacement Missiles and collecting tritium, since you have chosen an asteroid that spins very slowly it will be quick and easy to deplete the Tritium Subsurface Deposit.
Once the asteroid is depleted, return to the main game menu and reconnect, you will find yourself at a safe distance from the asteroids, do NOT move, look just below you and you will find the same asteroid a short distance away waiting for you cast your Prospector Limpet and start mining again, as you are thousands of ly away from civilisation no space pirates will come to disturb you, this tried and tested technique allows you to collect Tritium quickly and easily, you'll use up a lot of Limpet this way because you'll have to cast new ones for every turn of mining, but the Tritium you collect will far exceed the Limpet you lost and you'll have to discard excess Limpet.
If you are lucky, as happened to me, and find a very slow spinning asteroid with 4 Tritium Subsurface Deposit a short distance from the centre of the Tritium hotspot, you can return to your fleet carrier, dump the Tritium and return to the hotspot, easily identify the special asteroid and start mining again, I mined about 5000T of Tritium this way.
Here is my build :
Minatore Solitario
I would like to point out that precisely mining Tritium only makes sense if you are lost in the galaxy with your fleetcarrier with an empty tank! But absolutely try NEVER to find yourself in such a situation!