Ah, the classic translation paradox, something
everybody struggles with. The mix sounds good in one environment but not another, and fixing it for an alternate environment makes it sound worse in the first.
It would seem to be an impossible mission, if not for the knowledge that professional recordists do somehow manage to consistently pull off this bit of magic, so you know it
can be done.
The basic problem in a nutshell is that your listening environment (your speakers + your room) is not sonically neutral. Consequently, many of your mix, EQ and effects decisions are being made to correct defects that aren't really in your song at all, but rather in your environment. Consequently, when you take the song to another environment those corrections are no longer appropriate.
The ultimate goal is therefore to create a sonically-neutral listening environment for yourself. Not an easy goal! In truth, it's quite impossible to achieve a perfectly neutral space outside of an anechoic chamber. But you
can make it better.
First, the easy part: your speakers. Recognize that any speakers with 6.5" woofers or smaller cannot represent the entire low end of the spectrum. (Tip: on those Alesis speakers, try stuffing socks into the ports - it will help.) Understand that your speakers cannot accurately reproduce frequencies below around 60-70 Hz (maybe higher with the socks installed), and you will have to use a spectrum analyzer to check levels down in that range.
Try to situate your speakers as far from any walls as possible. Worst-case scenario is having them right up against the wall behind them, which will
seriously fu, er, mess up the low end.
Do not add a subwoofer at this stage. It will cause more problems than it solves. Rely on a spectral display such as Voxengo SPAN to judge levels at the very low end, and err on the side of attenuating them. Remember, most playback systems have even less bass than your studio monitors, and they're not going to miss that nice 40Hz whump at all if you filter it out entirely. Many commercial recordings routinely high-pass the 2-bus around 30-40Hz.
Now the hard part: room acoustics. That's almost certainly what's messing you up more than anything,
way more significant than your modest speakers or the lack of a subwoofer holding you back. I'd suggest a visit to Ethan Winer's RealTraps site as a good starting point; it's a treasure-trove of practical advice and information.
Finally, don't forget to use reference recordings. Load a commercial track that's similar in style to what you're working on - import it right into your project - and compare its spectrum to yours. Don't try to match them exactly, because you can't and shouldn't. But if yours is
radically different, that can be a clue where to start EQing.