It depends upon what public records you want to access...
If you're after something like the owner of a phone number, you're probably out of luck - commercial numbers are technically public, but they're not well published online for reverse search for free. Unlisted numbers are, well, unlisted!
Real estate (property) info can often be searched for at your county (or sometimes city) via the relevant taxing authority (for example the assessor or clerk or recorder office). Unclaimed property can often be searched at the state treasurer's web site. Business registrations, filings, rules and license info can often be sought at the state level. Building codes may be accessed at the local county (or city) website. Recent police ("blotter") records can often be examined at the appropriate PD or Sheriff's website. Court records are sometimes accessible via the court website, laws through the legislature website, and so on... the ability to search and the length of history of the information may be limited.
In general, start with the most local authority that is likely to have your desired info (city, county, state then country), and go through their website for the most logical department, and then to that department's site looking for online records access. If you don't find it, there's usually an email address for the relevant official who will respond with access information (if any) or hopefully a pointer on where to continue your search.
Usually the companies that sell such information are actually selling a service by indexing the information which is otherwise somewhat of a pain to access. Until recently, the information wasn't often online at all (the service companies employed people to enter the info into their private database).