Can I get internal US Flights for cheap when booked as part of an international return flight?

Several airlines did do that type of pricing in the past. In fact, I once flew from San Francisco to Chicago to Miami to New York and back to San Francisco onto an international trip for less than $100 total (most of which was taxes). However I am not aware of any US airlines that have such pricing now days - at least not on cheaper fares (eg, you will still find it as a part of a round-the-world fare, but not standard discount round-trip fares).

That said, you can still sometimes benefit from booking your domestic flights on the same ticket as you international flights.

Airfares are complex beast, but for most carriers now days flights are based on an Origin, a Destination, and potentially one or more Stopovers in between. Pricing for that part of the flight will depend on each of those things as well - and not necessarily in a way that might make sense to you.

So, say you wanted to fly from Amsterdam to New York, and then from New York to Chicago.

There's two ways that flight could be priced. One is as two separate flights (AMS-NYC, NYC->CHI). The second is as a SINGLE flight, with a stopover (AMS-CHI, with a stopover in NYC). Which of those will work out cheaper will depend on the airfares at the time you're booking - it might be cheaper to price it as a single flight with a stopover, or as two flights.

Now, lets say AMS-NYC was going to cost you $500. And AMS-CHI, with a stopover in NYC, costs $550, whilst a separate NYC-CHI flight might cost $150.

By booking the flights together, and getting it to price as a single trip with a connection, you've saved $100. Sometimes the price can even go DOWN when you add the extra leg - it might cost $500 to fly to NYC, but only $400 for AMS-CHI (even with the stopover).

Of course, it's just as likely that it'll cost more to book it as a stopover. In this case good booking engines will give you the cheapest combination of flights (one trip with a stopover, or two trips) - but sometimes the airlines will put rules in place to stop them doing this so it's always worth checking both options.

The other advantage of booking both the international and domestic flights together is that you'll get the international baggage allowance (which is normally better than the domestic one) for the entire trip. This can save a significant amount in baggage fees depending on the airlines involved.

Times have changed indeed.

There a few remnant programs left in Asia (Asean Air Pass) and Europe (one of the sky Team guys, I think) but to the best of my knowledge, not in the US.

1. You can typically add one domestic leg to an intercontinental trip (round trip or open jaw) for free or for little extra money
2. Any 4th leg is likely to explode the price if a single itinerary, so your are better off booking this as individual legs are separate mini itineraries.
3. The budget airlines (Jet Blue, Southwest, Virgin America, Spirit, etc.) tend to have a much simpler fare structure: one ways typically don't incur any extra charge. Fare structure for the mainline carriers (United, America, Delta) is insanely complicated.

These are all "rough" guidelines, air ticket pricing has become beyond complicated. Your options are

1. Book it all yourself. There are some really good tools for research (http://matrix.itasoftware.com/, https://www.flightconnections.com/, https://www.skyscanner.com/airline/airline-direct-air-d1.html) and booking (www.hipmunk.com, google flights, every air line website)
2. If you do it yourself, you can often safe a lot of money by jiggling dates and airports a bit. Many NFL stadiums have multiple airports in within driving distance. Tools like ITA Matrxi allows searching multiple airport, but it does take a bit of time.
3. Talk to a local travel agent: I would NOT recommend this. Most travel agents these days just don't have the skills and tools to do this efficiently and they need to get paid.
4. Talk to an agency that specializes in multi-stop trips. Example airtreks.com

No endors*m*nt intended or implied.