If you have an old, shitty battery, it can require more amperage/time to recharge than a newer, good battery.
You also have to be careful when switching to a deep cycle battery because they provide less cold cranking amps than a regular battery. If you live in a cold climate I would suggest sticking with a normal battery (a good choice would be a red top optima battery).
A deep cycle battery will provide more reserve power, but in my opinion, it is unnecessary unless you are listening to your system a lot with the engine off. And again, I would be hesitant to completely replace your normal battery with a deep cycle battery...most people usually add in a deep cycle battery to their system to compliment their existing normal battery. If you do this, however, you must be careful to ensure that both batteries are in similar condition and you have an isolator between them.
To be honest with you, your system isn't something that would warrant either a deep cycle battery or an additional battery. Simply having one good regular battery and a cap should solve your dimming issues, and is probably the most cost effective way of solving the issue.
If you choose to upgrade your system in the future (eg adding more amps) then I would suggest as big three upgrade as well as an alternator upgrade.
Also, a 6" speaker is not a sub, it is a midrange driver...and will not provide "more bump." It would be more beneficial to tweak your existing system...eg switching to a ported box for your subs or getting a more powerful amp, or if your subs and amp are compatible, rewiring them to run at a lower resistance (eg 1 or 2 ohm if they aren't running at those resistances yet).
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Although my battery is needing to be changed, before i hooked the inside speakers up, i was able to listen to music non-stop with the subs up very loud for 20-25 minutes w/o it dying,
After reading this, it seems you have let your battery drain numerous times. When you do this to a normal car battery, you cause irreversible damage to the battery. This is probably why your headlights are dimming because your current battery is so damaged that the alternator has to put out more amperage to keep it charged and the battery cannot do a good job of assisting in voltage drops.
If you like to listen to music with the engine off to the point of draining your battery, then I would take Subwoofer's advice and switch to a deep cycle battery. But again, you have to be cautious when doing this because it will not be as good as a normal battery for starting your car.