As I understand though, are rates are basically already at 0%. So bond rates cannot fall; at best they can stay the same, and at worst and most likely, interest rates will rise. So bond prices have no place to go but stay flat or drop. If I am worried about volatility, should I not just keep some of my investments in cash?
This is sort of a complex discussion.
Short dated bond yields are close to 0%, yes. However, if rates rise on those bonds, you wont really lose any money because the duration is so low. You can think of these bonds as slightly enhanced cash (they might yield 0.30% or so vs your checking account's 0.05%). You're neither going to make or lose much money on them.
Longer dated bonds have higher yields. Right now the 10y US government bond yields 2.75% which should at least keep up with inflation. These have some downside if rates continue to rise. For every 1% move in the interest rate, these should fall about 8% in price. The 10y has historically yielded about 2% more than inflation, so if rates entirely normalize over the next year, you'll probably lose ~14%. If we hit another rough patch (
http://www.bloomberg.com/news/2014-01-02/china-s-runaway-train-is-running-out-of-track.html), you'll probably make ~12%.
If you wade into corporate bonds, you can get even higher yields, usually on the order of 4-6% since you're being compensated for corporate credit risk. These bonds will have less upside if the economy struggles since they're not a "flight to safety" asset, and widening credit spreads could mute returns.
High yield bonds will often yield 6%+ due to their increased default risk. These have considerable downside in an economic collapse. See the ETF HYG or JNK to see how they behave.
If you want to roll up your sleeves, you may be able to find low risk bonds that yield 7-9%. I've given a couple examples so far in the "Trade Idea Thread". This process is probably too advanced for most investors.