The amount of cards given in a starter pack varies from one TCG to another. Starter Packs typically include 1-3 cards from the player's first choice deck and 3-5 randoms.
A player may receive a bonus to their starter pack if they join the TCG during its prejoin period.
Most TCGs are now automated and starter packs are usually generated by a management script once a player submits their join form. In the past all emails were managed through email so at times it took time for members to receive their packs.
Some TCGs will ask for players to wait to play games until after they have been officially approved, but not all of them.