Most pedal/crank combinations use one size of spindle and thread. There are a few random oddballs, but it's not incredibly likely that you have one of those. You do have to be careful though as one of those oddballs is very close to the standard size, and you can ruin your cranks if you try to force it. As Blam suggests, your best bet is to take it to a bike shop and ask them what you have. It's not something that anyone is likely to be able to evaluate online.
There are a number of different shoes that are clip-ins and are also (supposedly) suitable for everyday wear. None of them are really as comfortable as an honest-to-god sneaker since there's a chunk of metal screwed into a plastic plate underneath the ball of your foot. But they're better than straight-up road cycling shoes or even mountain bike shoes.
When shopping for shoes, look for something that, at the very least, has the cleat attachment recessed in the shoe. If it's marketed as something you can walk in, you're probably good. And as a general guideline, the more it looks like a regular shoe, the better it will be.
When shopping for pedals, look for something with a small cleat. The smaller, the better. Bigger cleats are more likely to protrude out from the recession in the shoe's sole. If it protrudes a tiny bit, it'll click with each step on anything other than carpet. And if it's really big, it'll poke out from the sole of the shoe and it'll feel like you're stepping on a rock with each step.
Specific product recommendations are off topic for this site, so I won't go into that part of your question. Just try on as many as you can and pick the one that's the most comfortable. Be careful if you try to attach the cleats to them. The hardware can cut into the mounting plate and then you wouldn't be able to return them if you didn't like them.
Also, check out the related questions to the right. You'll probably find some useful information there.