I want to upgrade my bottom bracket (square taper 68x113 threaded) to Shimano hollowtech and fit a new triple crankset. My gears are 3x10 so will my front gears still work with the new set up. What type of Shimano bottom bracket do I order.
Assuming your current setup is also 10s, it should work just fine. You need the spacing between the chainrings to be the same between the old and new cranksets. Otherwise, your front shifting is not going to work well.
If this is not the case, you will want to change the shifter and derailleur to allow you to have correct shifting.
With a square taper setup, you’d have to select the correct BB spindle length if you’re fitting a new triple. That’s often specified by the crank manufacturer. With the Hollowtech II cranks and other similar designs, the spindle is bonded to one of the crankarms. I believe Park Tools calls them thru-spindle cranks. The BB cups allow the spindle to pass right through them. Hence, you would use the same cups for double and triple cranks.
One point you may not have considered is the rear derailleur capacity. Capacity influences how large a cog you can run, and it is related to how much chain slack an RD can take up. The bigger the difference in your front chainrings, the less the spread at the rear can be. Officially, the current 10s Shimano Tiagra RD is rated for a 32t max cog if used with a triple (presumably a 50/39/30 Sora or Tiagra crankset).
If you wish to do the calculations yourself, find the RD capacity. The Tiagra unit I linked has a capacity of 41t. Find the difference between large and small chainrings. Find the difference between the largest and smallest rear cog. Add those two quantities. It must be no more than the capacity. (Or rather, it can exceed the capacity by a bit, but things are no longer guaranteed to work perfectly). You can see that (50-30) in front and (32-11) in back will produce 20+21=41.