There are 2 common ways to access Usenet groups today: connecting to a Usenet server via NNTP, and browsing through Google Groups.
There are a handful of companies that offer Usenet access. One of the most popular is news.individual.net (they do an excellent job of filtering out SPAM posts). giganew.com and supernews.com are also popular providers.
Google Groups can be accessed at groups.google.com. On that site, enter the name of the newsgroup you want to browse and click 'Submit' and you will be taken to a page with recent posts from that group. You will also be able to post messages to those groups.