Harrogate itself is a very nice place (very popular with rather posh tourists), and has high rents as a consequence. You might find something nice but less expansive further out, but that said North Yorkshire is expensive anyway. Rents might be cheaper closer to Leeds or Bradford, but you have to consider that there would be commuting costs and time to factor in. It may not look far on the map, but the traffic will probably be a lot heavier than you might expect. Also that the road may well be narrower and with more curves than you may be used to (although that depends on where you're from)
I'm not sure what Skipton prices would be like, but they might be a bit cheaper.
One other factor is that you will pay a lot more for detached than for a semi-detached, and even less for a terrace. Depending on where you are a lot of the less expensive housing will be terraces, and this is the case in a lot of parts of Yorkshire, and some of the Victorian era houses can be quite large. Don't rule them out- I live in a terraced house near Newcastle (so about 1 1/2 hours to the north of you) and it suits us very nicely (two adults, two children, four bed terrace with two of the rooms in what had been the attic). That said, no matter what you rent you won't get as many bathrooms as US houses, and even with four or five bedrooms and they will be smaller than you are used to. Gardens are smaller too.
Whether you get a basement or not depends on the region and how high the water table is! We don't have one here, but when my DH lived in Huddersfield he did have one.
I should mention that it is a lovely part of the country. The tourists go there for very good reason. You are within easy distance of Holmfirth where Last of the Summer Wine was set, and Thirsk, where James Herriot lived. Within about an hour's drive you've three cities, some amazing old buildings, fantastic museums and galleries, and stunning scenery. Most museums and many art galleries in the UK are child friendly, and some of them are free or offer season tickets.
You also mentioned homeschooling- there will almost certainly be some homeschooling groups in your area, and some of the local museums and art galleries may have good resources. Whilst homeschooling is not common in the UK, there are people out there who do it.
You mentioned libraries and playgrounds- I've had a quick look and there is one in Harrogate and another in Skipton. This link is to the local council website http://www.northyorks.gov.uk
with info on libraries and other local services. Don't forget that although you're in the middle of the country, you're still not that far away from the coast, and that can be a great day out- if you can get used to the idea of wearing coats and wellies on the beach! The nearest one will probably be Bridlington, but Whitby and Scarborough are also options. Whitby itself is worth a visit anyway- lovely place.
How much you have to pay for utilities depends a lot on what sort of house you have and what fuel you use. A well insulated property will save you a lot, and with a terrace you only have heat escaping through two of the walls rather than four. Gas heating is the cheapest, fuel oil is more expensive and electric is the most expensive. The only people I know who have air con in their houses are professional computer geeks who have to keep their equipment cool, and even then only in that room. Not that you really need it here...
I have gas heating and pay £140 per month for gas and electricity, but I do live in a very well insulated house. I have friends who pay a lot more. I can give you more of a breakdown of expenses if you want as there are local taxes, water charges etc to pay as well.