Why don't downstream outlets work after replacing a GFCI outlet?
First, identify LINE and LOAD on the device

It sounds like you got nailed by the positioning issue. There is no central authority who makes terminal positions exactly the same on every device.  If you want an eye-opener on that, just try a 3-way switch, which are different even within the same brand/model!   So position cannot be relied on; you must examine labeling and screw colors. 

On a receptacle screw colors will designate hot vs neutral.  (Silver=neutral=white).   Labeling and the warning sticker will designate LINE vs LOAD.  

You are also trying to throw all the wires on at once and go Voilà.  Even I don't do that.  Here's what I do. 

Also, no "back stab" connections.  This is where you jab a wire in a hole and the hole grabs it and holds it.   There's a different type called a "screw-to-clamp" where the hole is directly behind the screw and you must tighten the screw to clamp it; those are fine but tighten very firmly, preferably use the mandatory torque screwdriver to spec. 

If you are in the habit of cutting wire ends off, stop - length is too precious and we will be setting wires several times potentially.  

This will require a fair bit of capping off wires and testing, and many trips back and forth to the breaker to shut the circuit off while changing wires.  We certainly don't work "live". 

Hook up LINE, part 1

First, you identify the supply cable (hot and neutral) and hook that up to LINE.  Test that by hooking it to LINE and capping off everything else, then temporarily mounting it so you can push TEST and RESET without getting nailed by current.   Plug a load into the socket, a lamp will do.  

Power up the breaker.  Then confirm that the lamp lights, and the TEST and RESET buttons work appropriately.    If not, that cable was not supply, try another. 

LINE, part 2

Now, look for any other cables you want on LINE (that is, that you don't want to provide GFCI protection to).  Go ahead and hook those to LINE also.  If your GFCI only has 2 screws and any number of backstabs, then use pigtails. 

It goes without saying that you are attaching white to silver screws, and black to brass screws.  Also, the LOAD terminals are unused at this point.  

Now, power up the circuit and conform that the lamp, TEST and RESET still work, and that you're powering up the right (to be unprotected) outlets.  

Feel free to put a GFCI tester in them; they shouldn't trip.  (If they trip a GFCI somewhere else, then this entire part of the circuit is already on that GFCI, and this GFCI here is totally redundant.  Get rid of it.) 

LOAD

What remains is branch(es) you want GFCI protected. Hook up one cable to the LOAD terminals. (If you have multiple cables to protect, wait. Just one for now.) 

Do the testing dance again.  And also, go to the to-be-protected outlets and try a GFCI tester there.  


If the outlet is grounded, the GFCI tester should trip. 
If the outlet is not grounded, the GFCI tester should not trip. 


If the test fails, you have a ground fault in the downline of the cable you just added.  It might be a plugged in appliance, or it might be in the wiring itself, such as a neutral screw grounding out, or a crossed neutral or hot with another circuit. It is unlikely to be the wire itself.  Work that problem until it is solved and the GFCI tests clean.  

Then if you have any other cables to add to LOAD, unhook the LOAD cable now attached and add the next, and test it standalone.   Once they all test out, you can either attach 2 via Screw-to-clamp (if equipped) or pigtail.