This data is available in a number of places (...of course, at Forman et fil) but it isn't broken out in this particular format.
We give you the first three months of the 2013 season (April, May, June).
In the AL, no starting staff has thrown down a truly great month (sub-3.00 ERA). The A's and the Astros have had the best ERA in a single month, both of these in June.
It's clear that if the O's are going to stay in the hunt, they're going to have to improve their starters' performance.
The reliever totals show only one consistent team--the Rangers. The Royals and the Twins have been getting better work than the Rangers,while the Blue Jays' pen had simply an eye-opening month of June--which helped fuel their eleven-game win streak.
The Rays had a tough couple of months from their relievers, buy they had a very solid June.
Over in the NL, the Pirates mixed and matched their starters in May and June, and parlayed that into two months of solid performance. Their division rivals--the Cards and Reds--have also had good work from their starts (though St. Louis had some backsliding in June.) The teams with the biggest brick walls in their rotations over the first three months are the Brewers, the Giants and the Padres.
The Dodgers' starting pitching is coming around, but their bullpen remained inconsistent; they'll need to solve that problem in order to push toward the top of the NL West.
The Braves just seem to figure out a way to get a solid pitching staff in place, year in and year out.
Best bullpen in baseball right now? By the ERA numbers, it's the Twins. That should indicate to you that we are thus far having a very strange year...