Undershirt/base layer
Good as a first layer in cold weather, a long-sleeved silk, cotton, wool or synthetic undershirt can be bought quite cheaply. You can also use it as a thin shirt on its own in hot weather.
Boxers/shorts/undergarments
Whichever you wear (some travellers don’t wear any!), look for fast drying fabrics and choose comfort over style. Nylon is not so good in hot weather as it doesn’t breathe and this can cause infections.
Cotton underwear is good, as are some of the new synthetics. Some travel underwear is designed to be washed in the sink in the evening and will be dry by the morning.
Choose darker colours (not white), beige or off-white are good, as white will eventually go that way anyway. You can buy good quality underwear anywhere in the world (cheaply too) so don’t take more than 2-3 pairs to start with.
When I am at home in the States or in Europe I wear cotton boxers but when I am travelling, especially in hot countries (or far from a laundry) I wear 
Magellan’s boxers (shown here) as they meet all of the above criteria.
Long (thermal) underwear/long johns
You can buy high-tech, lightweight (polypropelene) underwear (leggings) which are good for layering. Try to get some dual-use ones that can also be worn on their own as trousers.
These combined with a similar top can, in terms of weight, be as useful as an extra sweater or down jacket for keeping warm.
You can also get leggings made of very thin merino wool. Icebreaker make the Bodyfit 150 which is the lightest pure merino base layer available. They make thicker models but the 150 is their all-weather model.
You can buy them from Blacks in the U.K., although they have stockists worldwide.
Swimsuit/bathing suit
Shorts can sometimes be used as a swimsuit (for men) and vice versa. Those with pockets are better for use as shorts.
Be aware that, despite their infrequent use in the USA, Speedos are worn all over Europe and are mandatory in some swimming pools there, they just won’t let you in if you are wearing any kind of shorts. But you can buy them (and shorts) locally wherever you are.
For women, a one-piece bathing suit or tankini can also be adapted for day wear.
Socks
Don’t wear cotton socks, even if you do them at home and are happy with them. Far better to use one of the new synthetics in hot countries, such as CoolMax, Capilene, and PolarTec. Socks should ‘wick’ (draw) moisture away from your feet, something that cotton does very badly which can mean blisters, infections, and bad smells. Cotton is also slow to dry if you wash your socks (or get them wet).
Make sure they fit your feet correctly and avoid one-size-fits-all socks.
You must wear socks with boots but you don’t have to if you wear sandals or running shoes. Bear in mind, though, that if you choose not to, your shoes will eventually smell (very badly).
Also, the right socks can help you avoid blisters and will stop your shoes and sandals from getting and feeling sweaty. You can get very short socks for sandals which are cooler and look like you aren’t wearing any.
Choose off- or non-white colours. Remember that socks can also be bought anywhere in the world so only take 3-4 pairs to start with.
Thorlo or similar are good for hiking or walking. These are padded and are made of CoolMax. They come in a variety of thicknesses depending on how they are used. For general use (for example, walking around cities in hot weather), I wear 
Thorlo Light Hiking Mini Crew socks (above). I also have a few pairs of
Brasher socks, also made of CoolMax.
