DMMs won't do inductance and oftentimes have a more limited capacitance range than LCR meters. Also some LCR meters can test ESR which is useful for finding bad capacitors.
Actually, I happen to have a DMM that
does inductance - a Meterman 37XR, used when I'm out of quality meters...
In any event, it has a resolution of 1 µH, and is completely worthless for measuring anything below 1 mH.
Anyhow: You need a LCR meter to measure ESR, Q, impedance, etc. You could probably do these with a scope and function generator, but it'd be a whole lot more work and maths! Also, most LCR meters let you set the measurement frequency, and I don't think any DMMs do that.
I highly recommend one if you already have the basics.