IQ is Image Quality.
Don't worry too much about ISO until 800, your camera can cope just fine. After that it still copes well the noise is just a little more apparent.
Are you shooting RAW? If so you need to some sharpening when you get home, the fringing is (likely) caused by Chromatic aberration and can be corrected (to a degree) in post processing.
Have you got access to Light Room or the like? I'm sure the default software that comes with your camera will have some, most, of these features any way.
Image Quality, wow I probably should have worked that out! I thought it was going to be some super duper technical term!
good to know about ISO, I did go up to around 400 the other day in fairly dark conditions and was convinced the images would be shocking. Good to know I can push that up a fair bit.
The image above was shot in raw and did have a little sharpening but I don't like to go too far with it as I think it can sometimes ruin an image but maybe I didn't go far enough with this.
I don't have Light Room, I think I'm probably at far too amateur level to be investing in software like that just yet and would prefer to get some better kit before I go down that route (no external flash, tripod is a little sketchy etc).
I am just using the standard "View NX2" that came with the camera which has done me ok for now bearing in mind I don't have much experience with PP, I just make a few tweaks to see what I think brings out a good result.
I really wouldn't worry too much about graining. The D5100 handles noise excellently to the point where you'll get usable results up to 6400. ISO 200 or 400 will show almost no noise at all. It's good to keep ISO low but not if it's going to detriment your images.
IQ= Image Quality.
Is this the standard 70-300 Tamron or the VC model? If it's the cheaper standard one, you have to accept that it isn't the sharpest lens ever conceived but it is capable of delivering reasonable results with a bit of care. Stopping down to F/8 would be a big help as would getting the shutter speed even higher to make absolutely sure you're eliminating camera shake. As others have said, you'd probably find the lens sharper at 200mm than 300mm, just the nature of the beast but if you can't get closer, that's not much use!
The green and purple fringing is CA (chromatic abberation). Again things will improve at shorter focal lengths.
It is unfortunately just the standard Tamron so I do appreciate that its not always going to produce the best results.
I'll defiantly look to bring the aperture down by playing more with ISO. Good point about the shutter speed as I think the lens is known for suffering with shake due to a lack of VR.
Chromatic aberration is something is a term I have heard mentioned but I don't know a lot about it so will probably need to do some reading.
Really appreciate all the feedback so far, some really good information coming out here.