As a fresh mercenary wannabe unit, your best option is to pay for transit with your combat units as bulk cargo, mothballed for transit. You're probably going to be operating as an auxiliary to some House unit, occupying some less-useful corner of their base camp. They should normally give you sufficient time to unload and get your units combat-ready again before you're assigned to actual duties, especially for garrison contracts or pirate hunting operations.
If you're planning on making combat drops, travelling as cargo obviously won't work, and few independent dropship captains/owners would be willing to take that risk for less than you're making on the entire contract....which means you either need your own dropship or can't take combat drop missions from any party that can't/won't supply one for the mission.
My preference in the OP's situation would be something along the lines of a Seeker dropship. It's reasonably common, with numerous variants, some of which have been adapted to hold a lance of 'Mechs in addition to vehicles and useful quantities of supplies and other cargo (which you'll appreciate if/when you salvage several hundred tons of potentially repairable equipment, but don't have time/money to fix and sell it before the end of the contract). That's the kind of mix that suits sustained operations, rather than quick "smash and grab" raids, although it's a bit weak on armament if it gets attacked. Another plus is its speed, as it can keep pace with a Leopard, or keep its distance from a wing of heavy fighters sent to intercept it in orbit.
Another valid option would be a dedicated cargo hauler like a common Mule, taking mainly defensive or garrison contracts where you have time to unload safely before suddenly finding yourself on the front lines.
