I've heard that some/most US airlines charge for any checked luggage, even the first piece. Is that the case? Are there any that don't? What kind of charges would I expect per piece if that is the case?
Yes indeed, most US airlines charge for all check baggage. There are exceptions if you have "status" with either the airline themselves or with any of their affiliated airlines (eg, Star Alliance, OneWorld, etc), or if you're flying in First Class, or if you're connecting from an international flight, or if you have one of their pay-for credit cards, but for most occasional domestic travellers you'll be charged to check any bags.
The fees vary from airline to airline, but are generally fairly similar. International trips are higher, although most airlines still do not charge for (at least) the first bag for most international trips.
The fees are available on their airlines websites. eg, for both United Airlines and American Airlines it's $25 for the first checked bag, and $35 for a second checked bag for domestic flights, and whilst the first bag is free for international flights the second will cost you up to $100.
Virgin America depends on the type of ticket you have purchased, but is generally $25 for each of the first two check bags.
The main exception to the above is Southwest Airlines, who do not charge anything to check your first 2 bags (with standard weight limits/etc).
Some airlines, such as Spirit, charge different rates depending on whether you pay for your checked bags in advance ($28/$35 for domestic up to 40 pound/18kg) or at the airport ($38/$45). Mind you, Spirit also charge you between $20 and $45 (up to $100 starting in November!) for your carry-on bag, so avoiding them completely is generally the best option.