I use mechanical relays to switch AC power for pumps and similar AC motor loads using DC signals from Arduinos and similar.

I have very very little experience with electrical engineering (I'm in the Lego mindset of I plug things in according to things I find online until they work), but is there something I should be using instead with 120V ~3-5 A loads?

You could use an opto-coupled triac but honestly for a hobby project a relay is a lot simpler.

http://www.electronics-tutorials.ws/blog/optocoupler.html

IGBTs. Solid state, 0 crossover, lower voltage to drive. Similar failure mode, but many more switches.

They also have a very ohmic resistance, and very little interference between control and load signals (except when switching) that semiconductors lack.

Funny thing is that a lot of people don't know that relay contact composition is different (and is usually mentioned on the relay's datasheet) depending on the type of load (most especially resistive vs inductive). They then wonder why when they use a motor on a particular relay, even with a snubber, the contacts stop working.

Relays are great though for motor controllers, especially when you need higher amperages (and don't want the huge expense a mosfet solution can sometimes bring); an h-bridge is very simple to build, and tying an n-channel mosfet (of appropriate amperage capability) on the low-side between the relay(s) and ground, and you get easy PWM control (just don't switch the relay while PWM'ing of course).

RF grade relays will cost you an arm and a leg though.

To that point, another mistake a lot of amateurs make is that they buy expensive things on DigiKey without looking on eBay first.

Amateurs are often poorly positioned to diagnose problems stemming from floor sweepings, parts recovered from e-waste, cheap knockoffs, or outright counterfeits. eBay parts not in those categories are often free samples you can request yourself.

Paying a premium for a new part with predictable characteristics isn't always a mistake.

How else do you suggest they learn? China == cheap education.

Or at the local HAM swapfest.

I think a lot of ham radio gear just uses standard relays for switching RF signals. Apparently they work well enough in many applications.

As long as you are loading with a resonant load then yes, it can work. But as soon as you are in any way reflecting power back on the line or have an impedance mismatch (which will result in reflected power) you're in big trouble.

Another problem is if you use one to switch from say an antenna to a dummy load, you might find that you are leaking quite a bit of power onto the supposedly disconnected antenna link. RF is tricky, especially at higher power levels.

One trick you can use is to use only single contact relays and to bend open the contacts as far as they will go as long as they reliably close again when the relay is energized.

That way you minimize capacitive coupling between the contacts. DC relays are your best bet for a starting point because they already have larger gaps than most AC relays (to inhibit arcing).