I wouldn't knock the fact you're using 'cheap portables' too much. A lot can still be heard with inexpensive radios. If your digital radio (does it have a model #?) has SSB, it's probably adequate for SWLing (the DX-350 is better used for bandscanning, and MW DXing / listening).
All I use is digital portables, and I've logged a lot of stations with them. Here in the NW you can generally use more wire with digital portable SW radios than in other places, because our signal levels aren't as strong as they are back East where a wire can easily overload a portable radio's RF section. You might try adding some extra wire to the SSB radio's antenna jack (if it has one) and see if you get better SW reception. Try 30 or 40 ft of wire and see if that helps. I once had a 70 ft wire and it didn't overload my digital portables.
As for what you can expect to hear, you'll probably hear a lot more foreign language SW broadcasts than English language broadcasts here in the NW US. I receive a lot more Spanish broadcasts and Asian language stations than I do English ones.
As for SW broadcasts, late evenings you can hear Cuba and Brazil (Cuba in English on 6010 and several other frequencies, Brazil in Portuguese on 6180 and 11780 khz). Australia and New Zealand have strong signals on the 25 meter band (11945 khz for R. Australia and 11725 khz for RNZI) and the 21 meter band (13630 for R. Australia). Radio Rebelde (Cuba) broadcasts in Spanish on 5025 khz. China Radio International has a couple broadcasts in English that come in pretty well at night, as does North Korea.
When propagation is working right you can hear a lot of foreign language stations from Japan, China, SE Asia and South Asia on 41 meters and 31 meters in the early morning hours.
If you want to listen to the ham bands, the 20 meter band (14000-14350) and 17 meter bands (18068-18168) have a lot of activity in the late afternoon and early evening. 40 meters (7000 - 7300 khz) has a lot of activity during the night and early morning hours. Sometimes during the early morning you can hear the Indonesian ham pirates talking in Indonesian (and sometimes chanting) just below 7000 khz, or in the low reaches of the 40 meter ham band.
80 meters I've found more or less boring, and the higher ham bands (15, 12, 10 meters) are usually more or less dead.
I'm not that into utility or numbers listening, so the others here will have more to say about it than I can.
I've found the SW station lists on
www.eibispace.de and
www.short-wave.info very useful because those two sites are quite extensive when it comes to listing foreign language broadcasts (EiBi also has some of the main utility weather broadcasts listed) -- and most of what I hear seems to be in Spanish, Portuguese, or some Asian or African language.