Yes, dial-up internet is still available. The few existing dial-up ISP's are now offering a "Broadband" dial-up. It is essentially a smarter version of traditional dial up internet service. And with the exception of the performance increases described below, it is exactly like regular dial up access in all other respects.
High speed dial-up service is not so much fast as it is smart. Your service provider uses a combination of technologies to optimize your internet speed resulting in performance increases of up to 5 times over regular dial up. This is accomplished primarily by compressing certain types of files like text and graphics so they are more quickly transmitted, and by storing frequently used files locally so they don't have to be downloaded each time you visit a page. It can take some time for the ISP to properly optimize your connection so you can benefit most from this technology.
Generally $10.00 - $15.00 per month.
Some companies are now offering an independent service incorporating the same technologies to allow any regular dial-up to benefit from accelerated browsing or "broadband dial up" technology. This service typically costs $5.00 to $8.00 per month. This may be beneficial for those who don't wish to switch ISP's or don't have other options.
Be sure to evaluate all options before choosing an internet service provider.