The first problem is that you may be using the wrong cable and breakers. NEC calls for 2 20Amp small appliance circuits in the kitchen, to accomplish this you'll need to switch to 20Amp breakers and #12 wire.
The next problem. You'll have to pull new wire anyway, if you want to hook up GFCI receptacles. GFCI receptacles will not work properly with a shared neutral, you'll end up with nuisance tripping with a shared neutral. GFCI receptacles work by monitoring the balance between hot and neutral, so if the neutral is shared the GFCI will not work properly.
To wire up the kitchen properly, you'll have to pull 2 new 12/2 cables from the breaker to the kitchen (all #14 wire on that circuit will have to be replaced). Then install 2 20Amp breakers, to supply the kitchen. You'll install the GFCI's as the first receptacle on each circuit, which will protect all downstream receptacles.
You can share a neutral between 2 GFCI receptacles. The catch is you'll have to pigtail the neutral to the receptacles, not use the neutral from the load side of the first GFCI to feed the second.
So you should be able to do something like this...
But not like this...
You'll then be able to use the load side of each receptacle to feed other devices, like this.
You are correct that the red should be power to the light, except the other side of the switch should take off from the line in side of the GFCI, there is no requirement for lighting to be protected by GFCI, so do not connect to the load side. The GFCI tripped because there was a mismatch in the currents between power wire and the neutral wire attached to the GFCI.
This is based on my best guess based on the information you provided. To be certain, you should examine the wiring in the other switch box to ensure the black is unswitched power, white is neutral, and the red is connected to another wire leading to the light. The main uncertainty is how the other switch plays into this scenario, as there were not enough wires for a 3 way setup with power to an outlet as well.
Incidentally, I would suggest avoiding the push-in terminals if binding screw connections are available. Doing so is more work, but it provides a better more reliable connection, provided the binding screw is properly used.
Best Answer
Easy peasy. Most GFCI receptacles have a method to attach 2 wires to each LINE screw.
That is exactly what you do. The two black wires go on the brass LINE screw. The 1 white wire goes on the silver LINE screw. And you're done.