bowling.com has a range of shoes and ball manufacturers that they sell. As far as shoes, the entry level shoe should run you around $30, and would be fine for a casual bowler, and last a few years. They are built like sneakers, and usually look like them. Spending more can get you a leather shoe, or it can get you fancy removable and interchangable soles, but that's for an advanced bowler. I wouldn't go higher than entry level on your first pair of shoes. You'll get a better sliding surface than often beat up rental shoes, and you'll not be wearing other people's shoes. The downside to going online is that the shoes may not fit exactly according to the stated size, so make sure that the site that you order from will let you return the shoe if it doesn't fit. I haven't had to deal with bowling.com for returns, but I know one person that had good luck with their return policy for shoes.

As for a ball, unless you buy a lot of balls and know how you want it drilled, you are still going to have to see a pro shop about getting it fitted to your hand. I don't really know a way around that. As for the type of ball that you want, there are a lot of choices, but they break down into three categories: plastic/ polyester, urethane (I know, it's a type of plastic, but when a site or shop says plastic, they mean polyester), and resin.
the plastic balls are basic and are usually used as lane balls. They react the least with the lane surface. Pros use them for spare balls so they go straighter. You can get these to hook, but it requires a lot of effort. If you're buying a KISS ball or one with a skull in it, it's plastic. The most popular of these is the "white dot" from Columbia, which come in wild colors. The only benefit you get from having one of these over a lane ball is that it fits your hand and you don't have to hunt for a ball. That's a big benefit, and if you're a straight bowler, you'd want to get one of these. white dots should run you around $65
Urethane balls are the next step up. They react with the lane and hook. They can have asymmetric weights in them to improve the hook. This is probably the category that you'd want, looking for a first ball and wanting to throw a hook. The urethane balls are generally made to work under a wider variety of lane conditions.
Resin balls, also called reactive, are more specialized and more expensive, and are usually specific to a certain lane condition

Even within the urethane category, there are a lot of polymer blends that are used that are supposed to give different reactions, and they will also use different shapes of the weight inside to give you more spin (which also depends on having the ball drilled correctly to have that weight move right). Generally, if you stay at the low end of the price range for urethane balls, you are going to get the ones that aren't as specialized and are more suitable for a casual or league bowler who deals with a wide range of lane conditions. I'd expect to pay around $100-120 for a urethane ball. If you go to a site that has a lot of details on a ball, aim for medium oil as the recommended condition.

The companies change their models so frequently that it's hard to recommend a specific model. I liked a Storm Ultra Hot that I had, but they don't make that any more. I'll probably go with another low end Storm the next time I need to buy a ball. But I was given a low end ebonite urethane ball and it works just fine.