This is a simple problem that you will have to have patience with....

Your tank is not cycled. I assume you have a proper filter set up with
filter media to act as a biological filter.

First thing you do is a 75% water change - that will drop your Nitrite
and Nitrate levels (and NH4 if any). This will buy you time and lower
the stress on the fish. Measure the levels after you do this change and
in two days, do a 50% water change. Watch the levels again and if they
start to rise, do another 50% water change. Keep doing this until the
tank stabilizes (levels don't rise). No need to get the levels to zero,
just low enough to be safe. Your tank will cycle in 7-21 days - but it
is critical you have a filter media for bacteria colony to grow on.

JB wrote:
Have been getting spikes in my Nitrite, Nitrate, & Possibly Ammonia...
My test kit shows Nitrate near 40ppm, Nitrite lethally near 10ppm, and
I can't test for ammonia, but it smells strong in my tank...

It's a 10 gallon freshwater tank with (3) 2" adult Sword Tails, and
about 10 baby fry in a net... I just performed a full water change as
advised by PetCo, and cleaned out all the algae in the tank and things
were fine for about 5 days...

Also my Alkalinity (KH) is 40ppm and I know it's supposed to be 120-180
for freshwater fish, but the Pet store has no idea on how to raise
that...

I've used the AmQuel Plus that treats Nitrate, Nitrate & Ammonia for
the past 3 days and still it does not seem to be dropping anything....
Any advice would be greatly appreciated as I am hoping my fish don't
die...