The problem with this question in my opinion is that it is too broad. If you want good answers, you should separate it in several questions.
I'd start by first getting the fountain part done. Find appropriate fountain body and see what volume of water you'd need for it. This will probably give you a starting point for pump selection.
I can't recommend you a good pump, because I have no experience in that field, but this article seems like an interesting starting point. Basically, you'll need a low voltage DC pump.
While selecting a pump, you should be checking put what types of solar power sources are available in your area. After you research that, it would be a good time to post a question here about driving a specific pump with specific solar power source.
Next comes the assembly part. How difficult it is going to be will depend on previous steps. You could probably find a prefabricated pump controller which will most likely going to be just plug and play or you could be building your own. This is where it gets really complicated. If you are going to be making your own, you will need to make a circuit will require soldering skills and equipment. You will also need a circuit board which you can make yourself or get one from a factory (if you can find one which will take small orders in your area) or use some type of prototyping board. If you are going to be making it yourself, you'll probably need to ask several questions for that step alone, since making a controller will require some experience.
After that, if something goes wrong, comes the debugging part. If the design is good, most likely the only problem would be bad solder joint. A good multimeter will come in handy there.
I do know that this answer isn't very specific, but your question as I said is very very very broad so I'm posting this more to give you idea for size of your project, because what you are asking here is pretty much equivalent of asking "How do I make simple text editor with absolutely no programming experience?" on StackOverflow.
You mentioned voltage details in your comment. A good question when asking for that would be something like "I have a pump which needs XX volts and YYY milliamps. I have ZZZ, XXX, YYY solar panels available. How do I make a circuit which could drive the pump using one of them?" or something like that.