How can I extend my two-month visa in Thailand to 4-6 months?

Come to Chiang Mai and get a education Visa if you want a year in Thailand.

Do not discount getting a retirement visa. If you have an Australian bank account, and have at least the equivalent of bt800,000 in it, and get a health insurance policy that covers bt400,000 of inpatient coverage, and bt40,000 of outpatient coverage, you can get a type OA retirement visa very easily from the embassy in Australia. This is good for multi-entry for one year, and if you enter Thailand right before it expires you get another year. You don't need any Thai bank account to get it, just enough money in your Australian bank account, and health insurance good in Thailand. For what you want to do, with the length of stay varying depending on what you want to do for a few years, this is the best for you. If you originally plan to just stay a few months, but then your plans change once you are here you are covered and don't have to leave the country.

There's no way without leaving the country.

You can buy a 60 day tourist visa before you go to Thailand, and get stamped in for 60 days. You can then extend this for 30 days at immigration giving you 90 total.

You would then need to leave Thailand and either go to a consulate at a neighboring country and apply for and pay for another tourist visa, and do the 60+30 thing again, giving 6 months total, or

Just leave after the first 90 days, cross into another country and immediately return, giving you visa exempt when you return. This will give you 30 or 45 days depending on when you make your 2nd entry, which you can also extend for another 30 days. Giving you 5-5.5 months.