Correct way to add power plugs
The simplest answer here is to continue what you are doing, but use an appropriately rated power strip. Standard 110V wall outlets on ordinary branch circuits in a US household are rated for 15A service. You can easily find power strips that are rated to 15A and beyond. If you want to be extra safe, verify the amperage of the circuit breaker (it will be marked) and make sure all power strips are rated for that amperage. Then the breaker will act as a protection device for your power strip wires.

The daisy chaining of power strips is okay too, but you need to effectively treat the longest length from outlet to last plug as your "wire length." The ratings on the power strips might all be 15A, but if you chain 5 together, the combined length might exceed the gauge wire used internally. If you want to be safer, consider one very large power strip and then oversized, heavy gauge extension cords coming from that power strip to devices too far to reach directly to the strip.