Administrative and Legal
Again, I can't give anything but very general advice. I work in the UK. Formal contacts are almost unheard of here, I've never had one in my life. In the USA, contracts are virtually mandatory. Depending on where you live and work, you need to know about your legal rights, what you can and can't do and what your responsibilities are. Things get even more complicated if you are working for clients in other countries with different legal requirements. You should consider getting some professional indemnity insurance.
Establishing a fairly tight brief (and price) and agreeing it with the client before commencing work should make everything run more smoothly further on down the line. If it all needs to be drawn up as a formal contract, then that is what you should arrange, having taken appropriate professional advice.
For longer projects, it is best to opt for staged payments. Break the project down into concept, design, implementation, maintenance, or whatever, and bill each stage separately.
Sometimes an advance payment can be arranged on work you are about to do and is especially useful if you have to pay subcontractors - illustrators, programmers etc.
Think of the brief as the foundation of a design project. If it is built on sand, the whole thing could collapse with dire consequences for everyone. Building on a firm foundation will keep the project on good standing, and your reputation.