I recently purchased 100' of 14 gauge wire (includes ground wire) for some exterior landscape lights. However, because the wire is much thicker than the wire that came with the landscape light kit the pins on the light connectors will not penetrate the thicker wire. Because I had already submerged the wire in my yard I did not want to buy another type of wire and have to start over again. The service people instructed me to buy wire connectors and to splice the wire for each light I connected to it. Do I need to use three conenctors for each splice (1 for black, 1 for white and another for the ground)? I'm assuming I connect black to black (1 connector) and then white to white (another connector) but I'm not sure about the ground copper wire.

Is it low voltage, or 120 volt lighting? Does fixture have place for ground, green screw, or
Have a green wire, and is the wire solid or stranded?
Is fixture all plastic?

It is low voltage and uses 4W bulbs. Fixtures do not have a ground connection, nor a grenn screw or green wire. The 14 gauge wire I purchased is stranded (ground wire in the middle). Fixtures are all plastic

I would connect a foot or more of the wire that works with your lighting to each fixture.
Then I would not cut the wire anywhere, but strip 3/4" off fixture wires, then attach to your #14 by stretching the insulation and exposing conductor, wrap fixture wire around exposed wire a couple of times(Til stripped wire is wrapped around completely), solder, then use something like 5200, stagger the splices.
Since you have 3 conductors, you could have 1 common, and 2 light circuits, switched separately or together.

Is there something else I can do in place of soldering, like electrical tape? Or is that not recommended because this will be somewhat submerged? Also, what is meant by 5200? Please clarify "stagger the splices".

Look for direct burial rated wirenuts made by King, see:

10222 BLUE DIRECT BURIAL WIRENUTS "DRYCONN KING DIRECT BURY" OUTDOOR WEATHERPROOF SILICON FILLED WIRE CONNECTORS FOR: MIN 3 #20/ MAX 3 #16 (SOLD PER EACH IN PACKAGES OF 20)

Or any equal wirenut connector that is rated for direct burial.

5200 is a Marine Adhesive/Caulk/Sealant. Very weatherproof, and durable. I believe made by 3M.
Staggering the connectors, is done so if insulation lost, wires won't short.
I would use the burrial rated wirenuts.