From what I can tell, to start you'll need Wrestler cards across the three core types: Male, Female, Tag Team. Those seem like the minimum requirements to at least start playing. Manager cards may be optional.
Currently, the floor price for Common Wrestlers is pretty low. At the time of writing, each are going between 0.08 and 0.10 SWAP.HBD (roughly 8 to 10 cents USD).
If you have a spare $1, buying a few of these now could be all you need to get started. These ALPHA wresters are supposed to get more valuable over time too. Once BETA wrestlers are introduced, rewards from playing will favour the use of ALPHA wrestlers. (And so on and so forth for each release.)