I often use a Fluke 179 DMM for various tasks, but for just checking batteries it is way to expensive, a Fluke 115 would do for that. The 115 has a 6000 scale, i.e. you will get battery voltage with 3 decimal digits, but the DMM is missing the low current ranges and cannot be used to measure low modes current draw and standby current with.
I'll vouch for the Fluke 115 as being the least expensive multimeter you can trust out of the box. Or perhaps the 114 if you don't do any sort of current measuring with it. While the 115 doesn't have a milliamp range, it will measure below 1A, but the cutoff seems to be about 100ma. So it is not ideal for testing very low currents like you mentioned, and if you test led's, may not have enough voltage to light them brightly. It is after all an an electrician's meter, not an electronics oriented one. For that kind of use my 87V is brought out.
Most importantly, the min-max function, which also beeps audibly at changes of 1mv, has a very important function when working with li-ion. When a cell that is under discharge or charge, when you hit the steep charge/discharge knees, the meter starts to beep very fast - important for safety in case something goes wrong. You can also use it to know if you have suddenly dropped a cell, or if a cell that seems normal just goes berserk. With the audible beeps, you can catch this activity before it goes to the extremes. The beeps on the 11x series are a bit lower than say an 87V, as the piezo beeper is much smaller.
Also not mentioned about using cheap meters is their measuring speed, and some tendency for over and undershoot. The Flukes are plenty fast and are bang-on without nary an under or overshoot.
For me, working on li-ion without a quality meter is just asking for trouble. I also vote for HKJ's wonderful work to be a sticky!