Seeking visit to get familiar with country

Yes. You will just need to renew your visa at one of the immigration offices at the end of thirty days. They will ask for a bank statement, passport, and to fill out the visa form that you get at the immigration office. They are also sometimes asking for proof of onward travel.

Thanks, info I'm looking for is how to find an affordable rental for a period.

What is proof of onward travel??

Typically a plane ticket. Some people will buy a ticket before they go to immigration, print out the confirmation (in case they are asked for it), and then cancel the ticket.  Just put "proof of onward travel" into google. You'll get some advice there.

In regard to finding a long term rental, if you find something you like on AirBnB, for example, you can contact them to ask about longer stays. Once you contact them directly you can negotiate a longer stay.

You can rent a car and planning your stay at different places in Belize to see where you like the most. You can plan your last stay in islands and return the rental car before go to the Islands.

I've been here for 2 months and this I learned:

1) If you're on a budget and already know in which area you're setting base, search a rental in Facebook marketplace and local groups. CALL the owners, you'll find much better deals than AirBnB.

2) Plus side on AirBnB: it will give you a nice invoice that you can show upon arrival or when renewing the visa. Sometimes they'll ask where will you be staying and if you have a rental arrangement in place.

But don't trust all AirBnB hosts blindly, text them or call them before paying. I was burnt twice.

First a long term rental, it was a clear scam and had to go through AirBnB support to get my money back, and maybe worst, as the place didn't exist really, they let me stranded on the day I arrived, to a new country, in the middle of a pandemic, with a curfew starting very early (It has since been extended to 10pm now, luckily).
Had to rush into a hotel and pay their rates.

2nd time was a highly rated AirBnB host, but managing the property remotely, so he found out pretty late that the last guest had trashed the place and was unusable. At least this time he made a hotel reservation for me and offered that instead (it was a short term stay). So... Be careful using AirBnB.

3) car rentals are not cheap, Crystal being the cheapest. Neither company will put any effort in making a sale. I explained to them that their competition was offering the same for 5% less, and she was like "oh ok, good luck then". Rent for more than 3 days at a time and rates lower like crazy, I mean, from $90 a day to $27, for real, check crystal rentals website. Although, don't be fooled, GST sometimes it's not included on ads (+12.5%), neither is waivers nor insurance. Check your credit card company beforehand as probably paying with it you automatically receive insurance from them and can save on that.

If coming to the island, better ask around in forums here or whoever you get to know. Golf carts are freakingly expensive to buy, and to rent. BUT, they are pretty ubiquitous, there probably is someone with one or more extra carts collecting dust and willing to rent it monthly for far less than the rental companies. I got a monthly rental for usd$250, when the rental companies will  normally charge you usd$25 a day.

So, try to make connections soon and get local help for items and arrangements if you plan to stay for more than a month. Paying "tourist rates" for everything is unnecessary.