Is it like star ships where you should save and search to get the best one you can from the start, because you have to simply abandon your current freighter when getting a new one? Is that even an option? I've only looked at two, but they both had 13 slots and cost roughly the same amount. I'm also guessing there's no way to get free ones like you can with ships?
From the No Man's Sky Wiki:
Freighters have a maximum storage capacity of 48 slots and the first freighters encountered have 13 slots. Once a freighter is purchased, the freighters the player encounters from then on will have 2 additional storage slots and cost about 2.5 million credits more than the last until the cap is reached.
Another Wiki states similar information:
Freighters can be upgraded in the same general manner as individual ships, with each upgrade adding a slot or two to what your current freighter contains (cost per upgrade will be roughly 2.5 million units).
The first one will always have 13 slots and there isn't a free one to begin with. The next one you buy will have 1 to 2 more slots and this continues until the max is reached. The price goes up by about 2.5 million credits each time you buy a new freighter.