The Tescun PL-310 or PL310ET are about the best available at $50 or less. Buy a reel-up antenna like this and plug it into the antenna jack. Use the radio outdoors or with the radio indoors and the antenna outdoors for best reception.
No problem. Also, one thing to be aware of is that the best radio in the world is no good without a good antenna. The telescoping antenna on the radio is good for strong signals, but you'll want to look into a wire antenna for it. There's a lot of science and mathematical formulas to come up with the best length, but everything depends on your local environment. The best way to go about this is to get a long run of 18-20ga wire and try running it around.
If you don't have a wire or the 3.5mm adapter, Tecsun sells a spool antenna which can get you started. The 750 I mentioned has a speaker wire type connection on the back that lets you jam any random wire in, while the PL880 needs the 3.5mm adapter.
It sounds counter-intuitive, but the more noise you hear, the better the antenna. Tune to an empty channel, lower the squelch. As you move or adjust the antenna, the noise will increase. Once you get the most noise, you can set the squelch back or start tuning to see what you hear.
Tecsun PL-310EET is a decent starter radio for shortwave and works well with a short reel-up random wire antenna like Tecsun AN-03L and similar. I have tested these and have given them as gifts.