You've mashed together two different phrases in your question:
Pick it up
- to move, work, etc., at a faster rate.
And, as you show in your question "Pick up the pace"
to speed up the tempo; to increase the rate that something is being done.
These two phrases mean about the same thing.
It would not be acceptable to combine the two phrases the way you have:
*pick it up the pace (bad)
That being said, in your example, it would be understandable for you to use "pick up the pace":
You could start out with one training a week, and then pick up the pace to 3 times a week.
You're right that "pace" usually refers to the tempo or speed of something but it's acceptable to use it to increase frequency as well.
Here's a quote from an article:
In terms of future expansion, McVeigh said: “We did two last year. I’m hoping to do two this year and pick up the pace to three to four the following year. We would like to get to 10 by the end of 2014.”
If you're worried about this, you could also use "frequency":
You could start out with one training a week, and then increase your frequency to 3 times a week.