With the possible exception of Niagara (Niagara Falls is a smaller, more tourist-oriented city, so the public transportation may be somewhat less comprehensive), all of the cities you listed should have extensive public transportation coverage. Most cities offer some form of unlimited travel pass, and in most cases a weekly pass is available.
I think in general, you will find that the schedules of US public transportation will be a bit looser than you are used to in the UK (at least from my limited experience of the London Underground). Pickup times can vary by 15 minutes or more in many cities without being considered unusual, especially for the bus lines.
Generally speaking, you want to be a few minutes early (5 or more), just to be safe.
Philadelphia, as Raj More mentioned, is serviced by SEPTA, and has a good combination of buses, subways and trains. The trains are generally reliable, and provide fast and easy access through the main parts of the city, operating out of 3 main hubs within the city, and provide access to most of the suburban areas. The subways provide good access throughout the city, to the areas not immediately serviced by the trains. The buses provide access to just about everywhere else, although they tend to run less frequently, particularly for the less popular destinations, and their schedules are more... flexible.
Chicago is served by the MetraRail that is similar to Philadelphia's SEPTA train system, providing access to most suburban areas. There is also an above-ground elevated train that provides good access within the city. I am less familiar with the bus system, since during the time I lived there, I took the elevated train ("the El") when traveling within the city.
New York's subway is an excellent means of travel within the city.
Edit: I missed Las Vegas, which deserves some special mention.
Las Vegas may be a bit atypical, as, much like Niagara Falls, it is largely tourist-oriented. However, it diverges considerably from Niagara in attitude towards tourists. The lure of the free or cheap deal is frequently a means of attracting tourists to specific casinos, and this can frequently be exploited for traveling. Many of the hotels and casinos offer free shuttle services, plus there are a number of non-traditional public transportation methods available: link.