To do it incountry (assuming you're an american) you'll arrive either on a tourist visa or a visa exempt entry, then go to a 90day non-0 (>50) then a yearly extension of stay (>50) - believe me if you can get a non-0A before you arrive in Thailand.
Time is running short but if you can I would advise getting an non-0A from a consulate in america before you come. You'll get up to 2 years in Thailand without going thru all the hoops of getting a "non-0" and then an "extension of stay" inside Thailand.
So you arrived in Thailand on the 20th march, in your passport there will be an "admitted until" stamp 89 days after that. (17th June)
If you leave and return to Thailand in this period and want to keep your current your admitted until date valid you'll need to have obtained a reentry permit before leaving.
Here's a post from Jan on extension of stay - marriage. Probably best to go to your local immigration office and get their checklist as it will / may vary around the country.
- you're not / won't be applying for a visa - you're on an extension of stay - all you'd be doing is changing the reason of the extension from >50 (retirement) to marriage to a Thai.