Adding Chemicals When cycling your tank

Hi,

I started my marine aquarium just over one and a half months ago.
Being new to the whole game, there was a lot I should have first read
up on before being impulsive and just going ahead and doing stuff. I
guess I was in too much of a rush to get my first fish in. Ok, so now
I have burnt my butt and I need to sit on the blisters.

After the first week of running my tank, I checked the ammonia levels
and saw that they were relatively high. So I popped on down to my LFS
and purchased a Ammonia Remover. As far as I understand now, this
remover basically 'locks' the ammonia in the water.

As time has passed I decided to introduce fish into the tank. I added
a sharp nose puffer, dragon wrasse and a coral branded shrimp. The
shrimp didn't even last half an hour because the puffer killed him.
This again was because I trusted the guy at the LFS who new less than
me and should rather have been at home baking cakes than standing at
the store wasting my money. The puffer then also killed the dragon
wrasse three days later. (The puffer also died a few days later. Maybe
he ate too much???)

As you can imagine, my nitrate and nitrite levels are sky high now,
but the ammonia is reading zero. I then purchased a shrimp again, and
by some miracle he just keeps living, even with the water being in
such shambles.
I have now built a sump for the tank. In there I have a protein
skimmer in the one compartment, then in the next compartment I have a
biofilter and bio-balls, the water also passses through stuff (can't
remmeber the name off-hand) to control the nitrates (which is not
proving effective). Then in the third compartment the water is pumped
back into the tank. I have been adding Stress Zyme to the tank to try
and breed bacteria to sort out the nitrates but it is also not
working.

Please could someone out there give me some advice as to how I can fix
the mess up I have created.

Thanks in advance:
Jay