Single engine private pilot reported an airspace violation and an aircraft produced obstacle warning while attempting to fly under class B airspace.

Narrative

While transiting northbound travelling inbetween Baltimore inner harbor to the west and MTN to the east; I was under the Baltimore class B 1500 foot shelf. I started a slight climb partly to avoid towers to my left and partly in anticipation of moving under the Baltimore 2500 Class B shelf. As I climbed I noticed my climb rate increasing so I pushed the nose over but at XJ:48 I went into the 1500 B for about 15 seconds. I was back under the 1500 shelf at XJ:49. According to my track log from Foreflight; I reached 1540 feet at XJ:49.I was receiving audible obstruction warnings about the tall towers from my avionics at the same time I was concerned about the Class D space to my right and the class B to my left. In hindsight; I was probably task saturated (I was the only one on board) and I was flying in an area where I don't normally fly although I am casually familiar with the area.There was no traffic near me or above me according to my ADS-B system.I should have leveled off sooner. I also should have monitored Baltimore but I was listening to MTN Tower instead since they were more likely to have local traffic of concern to me. Lastly; I should have review the obstructions in the area (fairly flat but with several communication towers) so that I could have been prepared for the avionics warnings.