Basic pruninh question

[Mercedes]

I've got a shrub growing thickly against a fence but I wnat is it to be taller. So am I right in thinking what I do is cut off all the low branches on the side furthest
from the fence - if you see what I mean.

And I keep cutting off shoots that want to be wide rather than upright yes?

Good plan, but some shrubs will do as you wish and some may keel over with hard pruning. If you can't look it up then I'd just go for it and see what happens. Trim out the lower stems from central stems you wish to keep, to get the right clearance, then thin out to get the right sort of shape. Use nice sharp clean saccateurs (excuse my spelling).
This may induce vigorous growth and many new shoots which you will just have to thin out as and when.

If you don't like it as it is, nothing to lose?