When a horse and rider pair are having problems, where should you first look-to the rider being badly trained, or to the horse? Which is most likely to cause the most problems?

Really it can be either horse or rider. It can also be both. If the pair is not compatible, then they just won't work out. If it's just one or the other, usually it can be fixed. Retraining a horse is possible, to train out any bad habits. Teaching good horsemanship to the rider can also succeed, with determination.

I always try to think that there should be a connection with the pair in order to work to the best of both their abilities fully. dreamer would you say that the rider is a little more hardheaded than the horse? :D

More often than not its the rider, assuming its a trained horse. Once you get down to a green horse/trained rider or green horse/green rider its a harder call and could be either or both. This is why you want the horse to know as much or more than the rider. A well trained horse is less of a risk for a green rider, but a green horse can be dangerous.

Does the gender of the horse/rider make any difference? I mean, are mare's more docile and easier to train than a stallion or even a gelding? Sorry... may seem like a dumb question. heh. I'm sort of a horse noob.

I've often found males horses to be more docile than the females! It really depends on the rider and the horse and their dynamic. So many factors go in to it, it's impossible to say without knowing more of the story!

I would have to say that the females tend to have the mama instinct and that can make them more likely to be spirited I guess. So if after subsequent training there is no improvement, should they look at parting ways?

Does that mean that really you should try and get to know a horse before you make a purchase?

I'm not an expert, obviously, but I would answer yes to that question for myself. I wouldn't want to buy a horse until I'd spent at least a little time with it to get a sense of its personality. Same as getting a pet.

I am finding this more and more interesting. Can someone tell me what a green rider/horse is, little confused there. Thanks.