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Making all and each side mission. Even though I think you get a lot more experience by doing all of the main missions, so probably is faster doing 2 main-quest only runs rather than 1 long as hell walkthrough.
Never use gel, so all the extra ones you find give you XP.
Do all the little fetch quests you see.
Interact with EVERYTHING.
What he said. That almost implies going to the citadel everytime you do a minimally-relevant mission and explore every corner of it because there will always be a dumb mission that will give you some (although ridiculous) experience.
Which reminds me something. During missions, you can find datapads or computers with relevant information, which give you exp, too. The bad thing is that in this game you must be at 2 meters or less of the item to see it's something important, so you'll probably spend a lot of time exploring (which isn't bad).
I do every main mission and side mission(not the fetch quests, the actual quests). I talk with every squadmate, and I interact with most things in the world. I also have all the DLC's installed and with an import I always start at lvl 30.
I'm not sure why you cant get any highter TC, because I always end up in the mid 50's.