chlorine tablets or just plain old bleach

If you continue to use tablets, use trichlor only (if that's what you have been using) and not any product that has copper or any other metal listed as an ingredient. Also, do not use cal-hypo tablets as placing them in the same vessel as trichlor can have explosive results.

All forms of solid chlorine have undesirable side effects with long term use. For cal-hypo it's climbing calcium hardness. Eventually the CH will get very high and cause scaling and possibly damage to pumping/filtration equipment. For trichlor and dichlor, it's climbing CYA. Eventually, the CYA will get very high and make any amount of chlorine you add to the pool ineffective for sanitation.

If you want to learn more, read through Pool School, starting with ABCs of Pool Water Chemistry

Also, add your location such as TX to your profile and the details of your pool to your signature in a similar fashion as mine so folks trying to help your here can see what you're dealing with.
 
So, I am guessing that you recommend using just bleach in a bottle for maintaining a pools chlorine level? Is this correct?

Yep. Bleach in a bottle or liquid pool shock, which is just a higher concentration of bleach. A more convenient alternative is a salt water chlorine generator system. That has some trade offs as well but does not cause long term chemical problems like cal-hypo or trichlor.
 
Read through the articles Jason has recommended and I think it will become clear to you that some form of liquid chlorine will be the way to go. You are lucky that higher percentages of chlorine are readily available in most areas of TX 10 & 12.5% are at most pool stores and many big box stores. This usually will save you money over 8.24% household bleach.
 
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