If I install a GFCI on the first outlet in a circuit will the other receptacles on that breaker circuit have GFI protection?
Finding the first

There's really no substitute for opening boxes, unhooking wires and turning the circuit on and taking measurements.  

You open up your believed-closest box, and remove the wires you think are downstream (away from the panel).  Tape off the loose wires so they can't short against anything, and plug a 3-light tester or nightlight.   Then go turn the circuit on, and see what lost power.   If all the other receptacles lost power, you're in the right box.  Otherwise try another box. 

If the receptacle still has power, you've identified the supply wires.  Otherwise touch the loose wires with a non-contact voltage tester and see if you can find the lit-up hot, or just power down and try a different pair.  

Usually there are only 2 pairs.   If there are more than 2, you may need to disassemble a pigtail. This is also a good time to do housekeeping like make sure your wires are on screws (not unreliable backstabs), upgrade to the $3.00 receptacles instead of the 50 cent builder grade cheapies, and upgrade to 20A (T shaped neutral) sockets if the circuit is indeed 20A and you expect to use those. 

GFCI+receptacle devices are very confusing because they are doing two things. Let's break it up. 

What is a GFCI?

A GFCI is a black-box which inputs hot(s) and neutral on its "line" side ... and then it outputs protected hot and neutral on its "load" side.  



That's it.  To signify the protected side, I use brown and gray wire in conduit work, but you can do the same by tagging wires with brown and gray electrical tape.  It can be a circuit breaker, or a dedicated black box.  If that's hard to visualize, look at this.  



This is a GFCI device in its purest form: a "deadface" GFCI. You can only hook this up just like the diagram: supply wires to LINE and downline load(s) to LOAD (otherwise what's the point?)    

Since you have first location and supply wires identified, do exactly that. Pigtail the loads if necessary, though some GFCIs use screw-and-clamp to allow 2 wires on a screw. 

Be prepared to install a box extension if the box is too small to fit the GFCI device and the wires. 

How do you get sockets at the deadface location?  Well, heh heh, I don't seriously expect you to use a deadface when you could use a GFCI+receptacle... it's just better for illustration. But here's the gotcha:  Since the sockets are pre-wired to the LOAD side, it's possible to miswire a GFCI+receptacle, attaching power feed to the LOAD side - in which case the outlets will be energized but the GFCI will not protect.

How to test for GFCI functionality

Use a simple GFCI tester.  Most of them come as part of a 3-light tester, in which case get a simple one**. 



You just push the button, and it induces a hot-ground fault which should trip a GFCI protective device.  If it isn't obvious, put a "GFCI Protected" sticker on the receptacle. 



Now, this external tester will not work if the receptacle is not actually grounded.  (there is no way to induce a ground fault without a ground).  In that case, get a 3/2 prong "cheater" and extend the ground wire/tab with other wires, to reach a place with an actual ground. Plug the tester into that and it should work/trip. 



Once it passes, also put a "No Equipment Ground" sticker there. 



Now if this is a GFCI device and testing it also trips another one, then most likely it's fed off that other GFCI's LOAD outputs, which makes it redundant. Remove it and put it somewhere that matters. 



** Especially avoid the ones that have digital logic to "help make the lights make more sense":  you are much better off with a simple one and interpreting the lights yourself than using the silly legends.  The words describing the light combinations, I call them the "magic 8-ball". But the lights are useful.