On the runup, you should primarily look for significant differences in the cylinders when you do your mag checks. That will tell you if you have a fouled plug or other problem, and if so, on WHICH cylinder. So you should absolutely look at your tach AND check the JPI as well.
A point of note: Most people, upon getting an engine monitor, don't really know what "significant" means. I'd just look for an EGT rise on all cylinders. Note that it will take a few seconds. Set the tach and mixture as you normally do. Then when you drop a mag, watch the EGT bars - they should all go up within 5 seconds or so. Put it back to both, you'll see them go back down. Repeat for the other mag. This is not a substitue for the RPM drop - it's simply additional information.
Same thing in flight - the really nice feature of the instrument is that you can see all the cylinders and spot a difference early. You'll get a feel for which cylinder is hottest on the ground and in flight (they can be different) and the "normal" behavior of the engine. Then when something changes you'll know and can investigate.
I would take this with a good amount of caution. CHTs by nature have a high degree of variability. On a cold day in a descent they'll be low, on a hot day in the climb they'll be high. If you have a 6 cylinder engine, each of the 6 will be different. Between the Aztec and 310, I have 4 6-cylinder engines that are monitored, and the CHTs do vary significantly on all the engines. So, make sure you understand the difference between a deviation from "normal" and standard responses to changes in the environment.
What you should do is use the engine monitor to help you determine when you need to adjust your mixture to keep the CHTs happy. You didn't say what kind of airplane this was on. Lycoming Service Instruction 1094D states that you should keep CHTs below 400F for maximum service life, so that's what I do on the Aztec (this is harder to do in the summer even with cowl flaps, in the winter it's easy). On the 310 (which has Continentals), I aim for 380, because that's the number my Continental-flying brethren have said works well for them.
You will get a lot of benefit out of engine diagnostics, as you can now easily determine which plug is fouled (if one is), and a host of other issues that may crop up. It is worth every penny to be able to tell my mechanic "The spark plug connected to the left mag on the #2 cylinder on the right engine is fouled."
On runup, now that I have such toys, I look for the proper RPM drop and lack of roughness, but I also like to make sure the EGT's all rise somewhat consistently when on one mag. That's about all I'm doing differently on the runup, though those who've had them longer may have some other tricks.
That's what I look for, too (as stated above), and think that's the best use of it on the runup.
The other thing I've been doing is leaning in the climb, using a Deakin technique: Check EGT shortly after takeoff, and then simply keep leaning to that EGT (usually ~1320 for me). Depending on your technique for leaning for takeoff up there in the thin air, though, you may come up with a better technique that includes the ground leaning.
That technique works, but the biggest thing to keep an eye on in the climb are CHTs.
