In general, your workplace will set what is considered professional attire. Clothing like polos, jeans, or cargo pants may be allowed in some organizations, but not others; for what clothing is allowed specifically, ask someone in HR, or look in your employee handbook.
That aside, the best ways to stay warm don't have to affect the outermost layers of clothing. Undershirts, long underwear, and double-layer socks will provide plenty of comfort, without changing the outside layer of clothing. In fact, you may not have to buy a new wardrobe at all, as long as you choose the right underclothing.
Layers are awesome. You can wear a thin thermal undershirt, a tank top, a t-shirt, a blouse/shirt, and a vest or sweater, which provides five (or six, if you wear both the vest and sweater) layers of clothing. Thick, long underwear under trousers is also a good idea, and multiple pairs of socks will keep your feet warm. If the day grows warmer (or you know you will be in a warmer part of the office), you can remove some of the under layers without changing your appearance.
Remember that many thin layers always trumps a few thick layers. Also, if you keep your core warm, it's a lot easier to warm up your extremities.
When buying clothing, don't just buy what's on sale, or what looks good. Look at reviews online, and shop around for used clothing for a better deal. Cheap clothing is often made with a very low thread count, which may save money in the short term, but will also hold very little heat. I have a few cheap shirts that are great for the summer - they're so thin, wind goes straight through them. The thicker the fabric and the tighter the knit, the more heat the clothing will hold. Since you want to hide multiple layers, you should get loose, long-sleeved shirts, trousers that are not too tight, and shoes that will allow an extra layer of socks.
If possible, buy insulated shoes, or insulated inserts; otherwise, wear long wool socks. Man-made fleece works as well, but I've found that they wear out faster.
Choose materials wisely
Cotton is a great fabric for summer, but it doesn't hold heat well. The thick weave of polyester holds heat well - but don't get too warm, because it doesn't regulate very well. If you get warm enough to sweat, polyester will move from your best friend to your worst enemy in the blink of an eye. Don't wear it next to your skin. Wool has the same thick weave, but it has the added property of being able to wick moisture away from the skin; it works as a thick outer layer or as a thin inner layer. Silk doesn't hold heat as well, but it makes an excellent base-layer, especially if paired with polyester. It's a good idea to research what materials are available; new materials can make layers thinner and warmer, not to mention more comfortable. Gone are the days of itchy flannel!
Finally, there are other possibilities beyond clothing:
- Reusable heat packs are small, gel-filled packs that hold heat and slowly release it. A heat pack in your shirt or trouser pocket can substantially warm you up.
- Single-use heat packs are great if you don't have a way to heat the packs; they come in many shapes and sizes.
- Electric USB heaters may not travel around the office, but they will warm your hands for long hours of programming; a quick Google search reveals pocket warmers, gloves, hand warmers, shawls, slippers, mouse pad, keyboard wrist-rest, and more, all in a variety of styles. Just remember to unplug them before you leave your desk!
- Heated chair covers won't travel either, but they are more permanent, so at least you won't worry about pulling your computer off the desk when you walk away. As someone who has a heated seat in his car, I can attest to the fact that a heated seat can turn a freezing room into a sauna in seconds.
- Coffee (or tea, my drink of choice) is warm and warming; even if you don't drink coffee, you can hold a mug to warm up, and microwave it occasionally to add the heat back.
- Other heated things include heated keyboards, hand-warming USB mouse, pocket heaters, and even an overheated cell phone.
- Exercise helps get your blood moving and your muscles warmed up; a brisk walk will do wonders to help you warm up.