Any of the tech fabrics by the major brands will be fine.
For a bargain have a look at K-Mart or BigW, both sell tech sports wear, otherwise go and have a look at some of the outlet stores in Smith Street (closer to the freeway end), Nike, Brooks and Adidas all have stores there.
"There is a crack, a crack in everything. It's how the light gets in." ~ Leonard Cohen
Yeah, as another 'heavy sweater', i got rid of the cotton gear and bought a tech singlet or two...much better....i know you said about not breaking the budget, but I also got a pair of skins knee length pants...i feel a bit like a dill all dressed up and then moving at a speed best described as 'glacial', but it is so much more comfortable, especially in 35 degree brisbane heat...very good value IMO
I was a total T shirt guy until maybe 2 yrs ago - now its only tech materials- it makes a huge difference- although some fave shirts dont see much light of day anymore ( ah....slaves to fashion.....)
I ditto the choice of the Smith st outlets- plenty of stuff along there, esp out of date summer short sleeves........I was there Sunday..
Not that I need much incentive- but pulling on tech shirts and shorts is the start of the running buzz........( a little bit of that Superman suit routine I loved as a kid.... )