FIRST THINGS FIRST ...
Wicking - the ability to move moisture away from the skin and spread it across the surface of the fabric so it can escape more quickly.
Comfort - no one wants a hair shirt against their soft skin, so a soft, comfortable fabric is preferable.
Anti-odour - the best baselayers will resist odour for a day or more, the worst will pong unpleasantly after just a few hours' use.
Protection - although baselayers are often worn under other clothing, they can also become the outer layer, particularly in warm conditions, so sun protection may be a factor.
FIT AND CUT
Ideally your baselayer should fit closely enough to allow sweat to be soaked up easily. That means a snug fit will work best for most people. However, in really hot conditions, it may be more comfortable to opt for a slightly looser cut.
There are all sorts of designs on the market. Short or long sleeved. Zip-necked with a collar or crew cut. Sleeves can be rolled up for venting or rolled down for warmth or sun protection. A long zip can be opened for increased venting and a decent collar will help to protect the back of your neck from sunlight and chafing from other layers. It's up to you what you find most comfortable - and what your chosen activity, time of year and location is.
In all designs, check for flat-locked seams on the shoulders. They're flat and are far less likely to lead to discomfort under pack straps.
Check too for scratchy labels. Either cut them out carefully, or look for garments with heat transfers that are simply printed onto the fabric and side-step the issue neatly.
There's now a wide range of fabrics out there and of course they each have their own pros and cons. Here's what to be aware of...
The most commonly used synthetic fibre is polyester. It usually wicks thanks to the structure of the yarn, which has a different weave inside to outside helping to first move moisture then spread it out. It wicks very well and dries fast too. However, it doesn't have natural anti-pong properties, so it needs either an anti-microbial treatment, which will eventually wear out, or silver impregnation, which is permanent and effective against smells.
Another synthetic fabric which works differently from polyester, polypro' is hydrophobic meaning that it doesn't absorb moisture. This means it dries extremely fast and moves moisture away from your skin, or at least doesn't hold the moisture next to it. It can still get a bit smelly after a day on the hill and in hot conditions can feel a bit uncomfortable.
Merino is a feel-good fabric in several ways. It is natural and renewable - coming from merino sheep - and ultra-fine fibres sidestep the itching issue you get with coarser wool, making it comfortable next to the skin. It also has the handy property of feeling cool in warm conditions and warm in cold ones. Nice.
Merino works differently from synthetics. Rather than wicking moisture it can absorb lots of it whilst still remaining comfortable, keeping it away from your skin even when quite damp. That's great if you're a moderate to medium sweaty person, but if you sweat a lot, you may be better off with a fast-wicking polyester top to move moisture away. Merino also takes longer to dry than synthetics, but doesn't smell and can be used for multi-day trips. It is quite expensive though.
A few companies use a mix of merino wool and synthetics. The idea is that the merino sits close to the skin for comfort and anti-odour while an outer layer of, say, polyester, moves the moisture away and outwards. The best of both worlds - in theory.
The hybrids can work very well - and do seem to combine some of the positive properties of merino and synthetics. However, it is hard to find, plus using the wrong balance of wool to synthetic can give disappointing results.
IT'S YOUR CHOICE...
There's no absolute best choice, just what works best for you. Everything mentioned above will work far better than a cotton t-shirt, which is really an absolute no-no in the outdoors. In general, the thinner the fabric the faster it will wick.
DON'T MISS OUT THE BOTTOM BIT!
Why technical underwear? It's the same theory as baselayers. To stay comfortable, you need to move moisture away from the skin and while cotton may be fine for day-to-day wear, for active use it's hopeless, tending to hold moisture next to the skin. There is no point spending hundreds on your sophisticated layering system yet keeping the items closest to your skin wrapped up in cotton, holding moisture close to the skin and stopping what's on top from working properly.
What fabric? The same factors that apply to baselayer fabrics are equally applicable to both knickers / underpants and sports bras - more of which later - but with one essential extra requirement. Because of their location, it's very desirable to use a fabric that has some sort of natural or added anti-odour property. As described above, the main synthetic options are polyester and polypropolene, and on the natural front, merino wool is another option. Finally, silk works well too, however it's expensive and not particularly hardwearing.
What Design? For men, there is the obvious choice between a longer-cut trunk or boxer short style or a more brief-like design. Trunks are less likely to chafe during walking and a close-ish fitting trunk arguably offers the best combination of wicking and comfort. Baggy designs won't contact the skin as thoroughly, which will compromise wicking.
Women tend to have more choice of design, but the key in both cases is to avoid anything that feels like it might cut into you uncomfortably - top or bottom - but is snug enough to avoid sagginess and slumping.
If you're an active sort of lass, you probably already have a vast selection of sports bras, but here are some basic pointers.
Fit is crucial, so if you're in any doubt about your correct bra size, get yourself measured by an expert. You need to choose the level of support your bra offers according to the impact level of the activity. Walking is low impact, but running off road is high impact and requires more support to minimise breast movement.
Build type may be related to support. Compression-type bras pull the breasts fully in and are best suited to smaller cup sizes. Larger breasts are more suited to natural shaped bras and may incorporate underwiring to provide extra support.
Styling is another consideration. Some tops are designed so they can be worn both as under or outerwear and are generally more generously proportioned; others are more like a traditional bra. There are various back styles available too.
Fabrics often incorporate Lycra for a close, comfortable fit, but otherwise, look for technical synthetic fabrics for best wicking and moisture management performance. Bear in mind that with extended use, the Lycra component of the fabric will lose its support and the bra should be replaced.