First of all, you need to choose your classification tool carefully. TWINSPAN is the most commonly used, but also one of the oldest and it has been superseded by newer classifications.

Check out the attached paper for a idea of different classification (clustering) methods and the references to them. Don't worry too much about choosing statistically between them at this stage.

For free software, your best bet is to use R, but be warned, it has a steep learning curve. You'll need to learn to code in R and find all the additional plug-ins and add-ons that people have created. Other than that, I've only used MVSP, PC-ORD and CANOCO, none of which are free but all of which have demo versions you can try out.

But just google "free vegetation classification software" and see what you find.