What is the best way to advertise a new web hosting company? What should I plan on, as far as a budget, for advertising to get some customers?

What type of customers are you after? Local market or Global? What type of hosting are you selling?

try to find a niche and sell to that.

I don't have time to market to much localy so most of it will be done over the web. I will be a shared hosting company to start off and as I get more experience I will move into dedicated and rent some servers.

Try hosting a couple of sites for free in exchange for banners. That site might then recommend you to people looking for hosting.

When you start out and your adverts budget is low you cant beat recommendations to start your signups rolling. The first 20 customers is the hardest bit.

From my experience , I know that : Adverts only work for known companies. They should be rather used to remind the market that You exist and you are everywhere . It's more for brand recognition rather than brand introduction. Unless they already know you , you wont get as many people interested in clicking a Ad.
Opinions may differ . :o

I'd try Google Adwords and FindMyHosting.com. Worked well for us!

isn't google adwords mighty costly for for normal "webhosting" keywords ? or are you using some localised keywords ?