There are three standard ways to move *things* between staging systems. The best combination of methods depends on what you're trying to move. Don't forget to move your jump clones.
Capital pilots should see the wiki entry about moving safely.
JFs hold bulk items. TEST Alliance offers a service that will do this stuff for you. See this forum post and the TEST Freight Calculator. Unlisted routes are probably fine. Try to use common sense for which calculator rate you should use or ask someone. Common intra-null routes are typically easier than high sec routes. Private services can often be found on the forums under "Services Offered". Those private services may have slightly different rules, but these are a good guideline.
Most capital ships have a Ship Maintenance Bay (SMB) capable of carrying fitted ships. Dreads, Carriers, and FAXes have an SMB size of one million m3. In this era, SMB is *usually* the preferred method of movement. For these calculations you want to look at the size showing in your ship hangar screenshot. Note that packaged hulls can not be shipped in this way and should instead be taken in a JF. Usually people will have extra space in their carriers to move extra ships. If you ask around in alliance, especially early and often in a move op, you'll usually find a couple hundred thousand m3 of space. Note that this method requires you to trust the pilot taking your ships. There's no collateral. You can also consider repackaging your ship and having the ship and modules shipped via Jump Freighter, though you'll lose the rigs. If you buy a ship off contract and the rigs are in the cargo, this is the reason why. The person selling those ships wants the ability to move them via JF without losing the rigs. If you've ever had more than 1-2b isk, you can optionally tip somewhere between 5 and 20m isk (rough, arguable guidelines). Maybe up to 100m isk if they're jumping twice solely for your benefit. No tip is generally expected. One null to null jump usually costs less than 20m isk in fuel. You're mostly tipping for the effort of clicking a few times.
Courier contracts should always have collateral. There are always spies watching alliance contracts. Either the stuff you're moving is cheap enough that collateral isn't a big deal or it's expensive enough to be worth stealing.
SMBs do require trust, but there are a few easy indicators you can use. You should always be calculating the value of your stuff vs the reputation/risk of the person carrying it for you. Look at how long the person carrying has been in corp. Do you know that they go on a lot of fleets? Do you think they value their position within the alliance more than the stuff they can steal? Just think about it and use some common sense. You get a little protection from the alliance, but this is EVE. Super pilots filling SMBs be especially careful when passing around other's ships.
Shuttles, those simple ships with no slots, are ideal travel ships now. They align quickly and are some of the few ships to have Interdiction Nullification built into their hull bonus after it was made an active module in 2021. They die instantly to smartbombs on the gate. With clever use of dscan and pings these can usually be avoided, but interceptors and yachts tend to be more durable.
https://wiki.pleaseignore.com/military:doctrines:travelceptor are a bit more durable and can carry a little more. They also warp significant faster. See https://wiki.pleaseignore.com/military:doctrines:travelceptor for more info.
At this point in TEST's history we have more than enough capitals and supers and JFs to get your stuff moved. You should really only be taking a titan bridge to move your pod, or maybe if you have some 10b blinged subcap that you really don't trust anyone else to take for you.
Another possibility is to extract via Pochven.
The idea is to use filaments to first jump into Pochven (a seperate kind of space only accessible via filament or wormholes) wait out your 15min timer there. Either cloaked or docked up in a blue structure and then use an extraction filament to go to a hi-sec or low-sec system.
https://pochven.electusmatari.com/ describes the possible target systems for each pochven system.
If you only have a pod in a shuttle to extract: Use a svarog filament to get into pochven. Dock up on a blue player structure in Raravoss or Skarkon till the timer is up Move to “Nani” and extract from there with a extraction filament. This will typically get you within a couple of jumps of Jita
