Fishkeeping - Adding Chemicals When cycling your tank

Jay, I'm by no means an expert, but I will offer you some advice based upon
my own experience:

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.

Ammonia removers have never worked for me, and I have never had anyone tell
me that such "removers" have worked for them. The best way to rid your tank
of ammonia in the short term is to perform water changes. The best way to
remove ammonia in the long term is to properly cycle your tank so that the
appropriate bacteria is in place to process the ammonia. There are also
bacteria additives that are effective in accelerating the process (by
introducing more bacteria into the tank). Of course, if there are no
fish/invertebrates/etc. living in the tank, don't try to lower ammonia
levels, as bacteria needs the ammonia to "feed" on in order to establish
itself in the tank.

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.

All of their days were numbered anyway, given the conditions of the tank,
especially since you mention nothing about changing out the water. Ammonia
and/or nitrite levels would have eventually grown to toxic levels without
frequent water changes (and maybe even with frequent water changes). In my
opinion you shouldn't consider adding fishes or invertebrates to the tank
unless the tank is fully cycled and your ammonia and nitrites read 0. Of
course some people use inexpensive fish to cycle their tanks (I use food).

wrasse three days later. (The puffer also died a few days later. Maybe
he ate too much???)

Read above...Probably toxic water conditions.

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.

As I mentioned before, you really should try to get your ammonia and
nitrites down to 0 (or close to it) before introducing
fishes/invertebrates/etc. into the tank. As it stands, if you want to keep
the shrimp alive, I'd suggest you do frequent water changes until your
nitrites and nitrates are under control and continue to do frequent water
changes until your tank is properly cycled. Your nitrate levels can be
somewhat higher, but water changes will help to lower them as well. Also,
when you finally see 0 ammonia, 0 nitrites, and some level of nitrates, it's
a
good indicator that your tank has cycled properly. If you're not up to
doing water changes daily or perhaps every other day until the tank has
cycled, you might want to take the critter back to the pet store.

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.

Stress Zyme isn't really for controlling nitrates. It introduces and
accelerates the growth of bacteria in the cycling process for your tank to
help reduce ammonia which subsequently leads to the growth of bacteria that
processes nitrites. Indirectly, I suppose, if you have anaerobic areas
within your tank (read: plenum or the like), it could help in the processing
of your nitrates by virtue of accelerating the growth of bacteria that
eventually leads to the creation of nitrates, but I suspect your
expectations are that it would help you get rid of the nitrates
directly...It won't.

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

I would suggest that you ask your LFS to hold your current living critter(s)
until your tank is properly cycled. It sounds like you're more than
half-way there anyway, so it shouldn't take more than two or three weeks to
finish cycling the tank. Barring that, you need to do frequent water
changes (daily or every other day, perhaps) to prevent poisoning your
shrimp. Test the water regularly until the tank is fully cycled in order to
establish how frequently you need to do water changes. Also keep in mind
that cycling the tank may take longer with frequent water changes, as you'll
be reducing the elements (namely ammonia and nitrites) which help to
establish or "feed" the necessary bacteria.

Also, rather than using a live creature to generate the ammonia which is
necessary to cycle your tank, you can regularly drop fish food into the tank
which, if uneaten, will turn into ammonia and be processed accordingly. Of
course, if you cycle your tank this way, you'll want to do a hefty water
changeout to lower nitrate levels before introducing fish/invertebrates/etc.
into the tank.

One last note: when doing water changes with live creatures already living
in the tank, make sure the salinity, PH, and temperature are the same in
both the replacement water and the tank.

In any case, good luck.

JBH

Thanks in advance:
Jay