Visit an antique store, garage sale or shop online at a website that sells antique furniture. Only purchase the furniture if you are able to inspect the piece. Even if the online description says that it is antique mahogany, you will need to get close to it to make the call.
Look at the color of the furniture. It should be deep reddish-brown.
Knock on the wood to determine its density. Practice identifying the sound of knocking on another piece of wood. The mahogany should have a deep sound, rather than one that is higher.
Look closely at some of the details. If the piece doesn't have a lot of details, it is likely not genuinely antique because antique furniture is hand-crafted with many details.