cerebral:
Thank you both very much. That is an awesome start for me. I just really needed to narrow the field on the relay/FET question to get started. Is there going to be a decisive advantage somewhere for logic level FET vs. transistors, or is it just matching maximum current?
Relays are mechanical things that go click, they're fairly easy to use and very robust. FETs are smaller, can switch very fast and therefore do PWM for brightness control, but have a habit of blowing up if maltreated slightly. BJTs are robust but have a big (~0.6V) voltage drop, so they tend to waste more power (get hotter) than a FET carrying the same current. Really, either will work fine for your application but FETs will make more sense if you need to do dimming and control more than about 3A. Large FETs and BJTs will both still require some support circuitry in order to provide enough current to their input (base/gate) - the BJT needs lots of current continuously just to turn on, and a FET needs a short but large spike of current in order to switch quickly and minimise self-heating while it's half-on.
cerebral:
The SMD3528 single chip LED strip I use is .08w per diode and typically I will rarely use more than 5 meters of it, or 300 diodes per picture (often well less, and occasionally I could have as few as 6 diodes). For the 24 watts or less per 5 meter strip, I buy the 3 amp 12V transformer . I may ultimately use a multi terminal, heavier duty transformer to run each "picture" to, but for now, they each have an individual 110/24V transformer. Thanks again for your help.
Don't buy parallel LED strips/panels, get series ones. You can buy LED power supplies from eBay for $5-$10 that have current regulation and will support any output voltage in some large range, e.g. 15-40V or 27-60V, see items #400687848188 or #400687964254 for example. That means you're dealing with much smaller currents, but you can't really do PWM because it will interfere with the current regulation in the power supply.
If you want to do PWM, you need a fixed-voltage (not fixed-current) supply. Again, you can happily get 40/60/72V supplies that will run really long chains of LEDs in series with not much current. You just need a transistor or FET that can handle the higher voltage (easy), but at far lower current than if you did them in parallel.
Beware of PWM on long chains of LEDs though, you're building a big antenna and driving it with a high power square wave. That means lots of EM interference out to many MHz unless you get the suppression right.