What are some useful (starter) metrics to use on phylogenetic trees?
Here are a few metrics you can calculate to get you started. You can use R to perform these calculations. 

Net Diversification Rate (r)

Net diversification rate is (rate of speciation - rate of extinction). You can calculate it using the bd.ms or bd.km functions in the geiger package for R.

r = 1:


r = 2:


Tree Imbalance: Colless index (I)

Colless's index measures how imbalanced the tree is. It sums the differences in the number of clades in each pair of taxa. You can calculate it using the colless function in the apTreeshape package for R. 

I = 0


I = 21


Branching Times (gamma)

The gamma statistic measures how much later or earlier branching times happen than what you would expect from a normal birth-death process. A negative value means branching times occur earlier, positive means branching times occur later. You can measure gamma (and generate a lineage through time (ltt) plot, as shown below) using the ltt function in the R package phytools

(From this Nature article)


These are just a few of many parameters you could calculate for a phylogenetic tree. It should be enough to get you started.