One of the larger Companies handling ferry journies across the Straits is :
ACCIONA TRASMEDITERRANEA
You can get an idea from their web site: http://www.trasmediterranea.es
You will be able to pull up a list of their particular sailings for the near future for both the fast 'Rapido' or 'Alta Velocidad' ferries (like a catamaran) or the slower car/lorry ferries.
I had a quick look and it seems that they offer online booking for the fast ferry and not the slow one. Maybe I missed it. The fast one currently appears to cost Euros 42 one way. I am sure the slow one is less but exactly what the cheapest fare is will need to be checked. Certainly the nearer you get to Africa, then the more 'negotiable' the cost. Will try to find the names of other ferry operators. ACCIONA is a well known company.   today, FRS-fast ferry mainly do the Service from Tarifa over to Tangier which is quick and by Catamaran. The prices are on the website.   here is a Credential email for a Private tours In Tangier Including fast ferry tickets/day trippers: FRS.reservation@gmail.com   I think they also do a cat service from Algeciras.
Algeciras has several ferry companies operating. The ferry port has several (mainly Moroccan run) sales outlets that can sell you tickets and you can look around. Or just look at the board showing next sailing inside the main departure hall in Algeciras ferry port and buy a ticket at the ticket booth.

Bear in mind that the Tarifa to Tangier ferry may take say 35 minutes to cross this shorter hop, and if you take a Conventional car / lorry ferry from Algeciras to Tangier it will take longer - say 2 hours and more, but on a nice day you can walk around on deck and enjoy the scenery. Maybe even see whales or dolphins!