Everyone's HR is higher running than cycling. There's more whole-body muscular activity when running and that increases HR at the same intensity, intensity being measured compared to LTHR. LTHR is higher running. Whether you get a power meter or go by HR, you have to go through the same process of examining your data to set training zones or targets. Your running experience is of little value for that, since your HR will be different and you don't measure power when running.
Personally, I've found HR to be a fine measure for incorporating varied training activities Though each different activity uses different HR zones, I can set up the various zones vs. activities in TrainingPeaks to give me comparable numbers for training stress. IME measuring training stress is the most help when trying to modulate training.
Joe Friel's Quick Guide to Setting Zones | TrainingPeaks
That said, I believe you can also figure training stress similarly between running and cycling by using power for cycling and pace for running, though running pace in hilly terrain will be very inaccurate, while HR will be quite accurate.
It can seem difficult to get one's HR up when one is very conditioned. I've found that it helps a great deal to do some early ride spin-ups/power intervals, say two 1-minute hard high cadence efforts with 5 minutes between.
Other commenters' admonishments about maintaining a high cycling cadence are well taken. 85-95 is usual for the flat and 75-85 usual when climbing. I'm not a tri guy, but I think 90 cadence is common for the bike leg.
More tri information in the Triathlon Forum.