Hello Conhall1958 --
Your being an American and wanting to become autónomo is really contingent on your visa status that you will have in the country.
Just because you come and sign up for a TEFL course -- that will not necessarily make you legal to be a self-employed worker which is what the "autónomo" status is.
To be autónomo, you need to have a NIE (número de identidad de extrajero) plus legal working/residency status in the country.
You don't need any start up capital since you are registering as a self-employed worker, not a company (sociedad). However, you will need to make minimum monthly payments to Seguridad Social of at least 250€ once you register.
When you invoice your clients, you will need to retain the IRPF (impuesto de la renta sobre personas físicas). This retention applies to invoices to businesses not individual people unless you are billing another autónomo.
You can read more about being autónomo here --> http://www.empleo.gob.es/es/sec_trabajo/autonomos/economia-soc/autonomos/index.htm