What are my options for extending my stay in Thailand beyond the initial 30 days on arrival?

Thailand is easy for visas and if you have enough earnings then long visa is easy. Already as tourist you can stay 90 day's a go out and return

Perhaps consider the METV (Multiple Entry Tourist Visa). It's perfect for stays for up to 180 days with unlimited exit/reentry and no interactions with immigration. You are required to exit prior to each 60 day stay to extend the visa for it's duration. I don't know your status but as an American you might find it prudent to consider being in the US for part of the year to use your Medicare coverage. You'll also find it easier and less costly to purchase travel insurance to cover you for the time you're in Thailand.

Apply for eVisa on-line, before leaving USA. This will get you 60 days, instead of 30 days Visa Exempt. Can get at least one extension in Thailand for 30 days. So 90 days EZ. More than that, you will need a nother Visa like the retirement Visa. Or fly out of Thailand (Cambodia, Vietnam...) and fly back.

Coming from the US, it's actually called "Visa Exempt" i.e. no Visa is required for 30 days.

That is completely different than "Visa on Arrival", that is for some other countries that are not eligible for "Visa Exempt". There are desks for "Visa on Arrival" before Immigration, however don't go there!

You can do the Retirement Visa, however there is a fair amount of money and requirements, but is doable. You can also apply to a Thai language school, however please register with one with the intent to actually attend classes. From my discussions with one school, the attendance requirements is not that difficult, and easily manageable. You can apply to a school after you arrive in Thailand, and they will manage all of the paperwork for you. They have 6 and 12 month programs available. Also some costs involved, but a lot less that the retirement visa, and a good option for your fist long term stay. Then if you still want to stay after this, can convert to the retirement visa.

The special tourist visa is 90 day and extended twice gives you 270 days.

Multiple Entry tourist visa.

mexico

You can convert to a. O visa based on retirement but you will have to lock up 0 baht in a Thai bank account for a few months and you will not be allowed to let that drop below 0 during the year if you think you would like to extend past a year. Alternatively, if you think you want to stay for just a year (or possibly two) but don't want to lock the money up, you can apply for an O-A visa in your home country. You just have to show you have the equivalent funds at hand but you can keep them wherever you like