If you are going to get a HAM radio with wideband receive go ahead and get your technician-class license. It is incredibly easy to get.
Simply by repeatedly practice-testing yourself (for free) from
http://www.qrz.com/p/testing.pl you will quickly gain the knowledge needed for the exam.
If you are willing to spend a little money the ARRL Technician study book
http://www.arrl.org is inexpensive, easy to read, and can probably be picked up at your local bookstore.
As for a small handheld HAM rig with shortwave coverage I have a Yaesu VX-3R and like it very much. It is absolutely tiny. It has a pretty capable litium-ion battery and can receive from 500khz to 999mhz (with some gaps) but all the big guys are covered: broadcast AM, broadcast FM, Shortwave, AM Ham bands (no SSB coverage), Aircraft band, Marine bands, and NOAA weather radio (with weather alert) and can transmit 1.5W on the 2 meter HAM band and 1W on the 440 HAM band. (power output is double when attached to external 12V power)
If you aren't planning on transmitting then a dedicated receiver is a better option. The Icom R20, though expensive and slightly large, would be your best bet as it can receive from 150KHZ to 3304MHZ continuously (less analog cellular) in all modes (AM, FM, Wide FM, USB, LSB, and CW. It also has dual VFOs meaning you can essentially monitor two frequencies at the same time. Also it can be computer-controlled.
If you are interested in HAM radio you might be interested in the Yaesu FT-817. This is considered a QRP (low power) rig. It has coverage for broadcast AM, broadcast FM, shortwave, all the HAM HF bands (HF:160M-10M, VHF:6M, 2M UHF:70CM) unfortunately no NOAA coverage. Plusses include very small (though not handheld), battery powered, great receiver Cons include mostly the fact that it's QRP. (5 watts max isn't much power output...I won't bore you with solar cycles and antennas and such)
http://www.universal-radio.com is a terrific place to window shop (and good guys to buy from)