I'd like to use it to experiment with and explore interesting parts of the spectrum. For example: decoding digital television signals, seeing what my cellphone sends/receives and how it encodes it, downloading weather satellite images, and so on.
I've no experience whatsoever in radio, only in software. Where should I start? What should I buy to begin with? Any general technical advice?
I've used the FunCube Pro+ dongle for amateur radio stuff. It's very narrow-band (it's based on a sound card as the ADC) but it does cover most of the important amateur radio bands (HF) which most SDRs you find on the Internet do not without additional hardware. It also has special filters around the 2m and 70cm amateur bands, the filters on 2m are especially nice because strong FM radio stations tend to desense the cheap receivers here (RTL-SDR, HackRF IIRC).
At work we use the USRP (B200) for WiFi-related testing, requiring the ability to receive and transmit above 5GHz over 80MHz of bandwidth. The RTL-SDR is not going to do that. (Actually the B200 doesn't quite cover 80MHz, but it's good enough.)
For actual amateur radio work I use an Elecraft KX3 with the 2m module. You can't beat the sensitivity of an entire system engineered to be sensitive. With off-the-shelf SDRs (the KX3 is SDR-based) there are going to be a lot of issues to debug to get the full system performing well. Fortunately, most of the hard parts have to do with transmitting at higher powers, so you might not care if you just want to listen.
A few friends have tried either the BladeRF or HackRF (I don't remember which) and find that it uses an IF near the FM broadcast band, and hence they couldn't get it to work reliably in Manhattan where they live because the IF stages are desensed by the super-strong FM stations nearby. Of course, debugging this is going to require already-working radios and test equipment (oscilloscope/spectrum analyzer). So these cheaper SDRs are not off-the-shelf instantly-working solutions. They're more of another thing that's helpful to have in your lab. You probably don't want to tie up your $10,000 FTDX9000MP to be a WSPR beacon, so you find a good-enough SDR instead and use that.
Finally, the good news is that most of the people writing software for the radio experimentation community know radio and not software, so there is plenty of room for improvement if you care to work on the software.