Trade is very simple, exchange of goods and services. Commerce is bigger, it's part of business, it has transportation, insurance, logistics, etc. Trade is a type of commercial activities. For example, when you go shopping and bought a shirt, you have traded your currency for goods. We can't say that with commerce. A commercial center (a shopping mall) is a center where many businesses practice trading and transport in/out their goods, they store them there or outside, they have to get a licence from a business authority (Chamber of Commerce), etc.

Trade can have a specific and technical meaning at the common level. Example: I give you my fish aquarium in trade for your book collection. This is also known as "barter" or "bartering". Otherwise, Trade is only another synonym for doing business, which is also termed "commerce". International Trade and International Commerce are terms that you will often see in essays about business.