I'd do some hard math first. Where will your downpayment come from? What closing costs can you expect? What kinds of costs are you going to have when you move in -- immediate fix-ups, new carpet, new blinds/drapes, etc.? How about moving costs? It looks like you have $25K of accessible cash. That is not a lot to cover all of the cash needs that come along with buying a new house -- at least, not if you want to have any kind of EF left.
I'm also going to assume that you want to rent your current house because you'd need to bring cash to the table to sell (you say it's worth what you owe -- does that mean mortgage balance, or mortgage balance + home improvement loan? In either case, selling means writing a check for closing costs, realtor fees, etc.). The problem is that, as others have said, renting is NOT a good idea. You will have maintenance costs, vacancies, periodic repairs/cleaning between tenants, costs in finding/vetting tenants, etc. -- my very first tenant went so far as to take all of the light bulbs with him! If you put a chunk of your $25K towards the new house, how much will be left? How many months could you carry the mortgage and bills if you had a vacancy?
The good news is that, unless your kid is developing slowly, early elementary school is basically irrelevant. If your kid is already reading, it is a waste of time, as the class will focus on the kids who aren't reading yet. Even if your kid is just learning letters and starting to sound a few things out, they will do just fine in even a bad school -- it's really the kids who are behind for whom early intervention is the most helpful.
So if I were you, I would have a two-year plan. Pay off the home improvement loan aggressively. Get the daycare costs down. Then use the cash that you free up to significantly build up your free cash. Then when you are ready to move, you will be at worst cash-neutral and may even have positive equity in the existing house that you can roll into the new house, and you will have plenty of other cash to cover the necessary expenses and still have a cushion left.
Tl;dr: You appear to have a history of letting your "need" for stuff get ahead of your ability to afford said stuff. As a result, you now have multiple debts that are stressing your budget and that have limited your ability to save the cash you must have for your newest "need" (the new house). The solution is not to take on even more debt so you can have what you want right now. It's time to break the cycle. Live on less, save more, figure out how to be happy with what you already have -- at least until you have sufficient cash in hand to afford the next step.