You need to consider the amount of power that could be dissipated in the LM317 and plan your thermal strategy to cope with that.
Basically speaking, the power dissipated is the difference between the input and output voltage multiplied by the output current.
So if you want 15V 1A from an input voltage of 19.5V then the power dissipated will be (19.5 - 15) * 1 = 4.5W. This will need a moderate sized heatsink. Something rated at 10 degrees C per Watt would be fine.
Heatsinks are rated by their increase in temperature for each watt of heat put into them. Larger heatsinks have a lower rise in temperature because they (amongst other things) have a larger surface area to radiate heat from. So 10 20degC/Watt heatsink is smaller than a 10degC/Watt heatsink, and a 4degC/Watt heatsink is much larger. This is a case where bigger is typically better.
The idea is to keep the temperature on the chip inside the device from exceeding its maximum temperature (preferably we try to keep it substantially cooler than this). A 10degC/W heatsink will increase in temperature by about 45 degrees and the chip inside the LM317 by maybe 10 degrees more. Assuming a max temperature of 30C, everything stays well below the max temperature (although you might burn yourself on the heatsink on a bad day).
However, the situation becomes far worse if you're operating the device at 2V and 1A. In this case the power dissipated rises to about 17.5W and a 10degC/Watt heatsink will increase in temperature by 175degC. Assume an ambient temperature of 30C and you are over 200C. You can check the specs if you wish, but this is going to be way over the max temp of the LM317.
Fortunately the LM317 will detect this and shut down, but it won't be doing it any good. Off the top of my head, I'd be looking at something rated at about 5 degrees C per watt, but things are still going to be getting quite warm.
There's lots more than this to thermal design, but this is a start.
Apart from that, Yes, an LM317 is ideal for this. It doesn't have a variable current limit, but that's generally more complex to build. Running it from a switchmode power supply means that the input ripple will be low, but also high in frequency. You may have to consider using additional relatively small value, but low ESR capacitors on the input and output to help reduce this.
Current is not supplied, but demanded. Your house is connected to a power supply capable of delivering at the very least several hundred amps (although there may well be a pole fuse that will blow as you approach this) yet your iPod charger (which requires milliamps is not overloaded. Why? Because it is the device which imposes a load that determines the current required. The power supply merely supplies the current demanded by the load (or at least up to the point that it can supply no more). When you switch on a toaster, the load your house represents becomes larger, and more current is supplied, but again, only as much goes to the iPod charger as required, and as much to the toaster as required. The same happens with any other power supply.