This is actually a tough one, normally speakers can make the biggest difference but you (generally you, not you personally) need a good foundation in a basicÿintegratedÿamp. A DAC would be a good purchase, but not yet, down the road a little when you want to get better quality once you have the basics taken care of - but you are already doing the correct thing by ripping your music losslessly for when the time comes to get the most out of a DAC. You are also correct in that the Bose (well Bose anything to be honest) aren't thought of terribly well, they are just marketed very well and are a name the average person knows about. Those Bose cube's are meant to be used with one of their sub's, but even then they are not what you'd call neutral or accurate.ÿ
You can go the Audioengine A2 route as they are aparently very good computer speakers, you wouldn't have to worry about an amp as they plug straight into the mains so take care of power on their own and plug into the computer via a mini-jack. The only issue here is that they are a bit limiting, you will not be able to plug your tv at the same time, nor upgrade the sound with an amp etc. If your computer and tv are near each other, you could have a lead coming out of both and switch the speakers back and forth, so if cost is absolute, then this could be an idea as you've point out.ÿ
For more options and better sound quality going forward, I'd personally try to stretch to the entry level Cambridge Audio 340A integrated amp which can be had for about £170 new. A great little amp that gives you a solid foundation going forward, and you can have the TV and computer plugged in at the same time obviously, in addition to a CD, DVD player or what have you depending on your needs. I'd then, depending on funds, either get Monitor Audio BR2's if your budget can stretch to them (just under £200 new) or beg, borrow steal and try to get some used Monitor Audio B2's, KEF iQ3 or iQ1's or Wharfedale 9.1 which are all very good and very reasonably priced speakers (my preference would be in that order, but any of them will give you great 'bang for the buck'). You can perhaps also save some money by shopping for the amp used too, if you can't find the 340A used there is also the possibility of a used NAD or Rotel int. amp, but the 340A would be my first choice in this price range and its the more expensive amps that tend to come up for sale used. Just my opinion, hope it helps a little in your choices as it is always a little too easy to spend other peoples money. You could get all the above new for under £400 and a great system to get your foot in the door and show the fuss people make about high end hifi, you're hear realÿinstrumentsÿand voices that sound like real voices. If you have to keep it closer to £100, perhaps the speakers first and then swap out the Teac next year when you have another £150. Good luck with your choices, remember, its meant to be fun so don't stress too muchÿ