Contracts keep both you and the client honest.
Pretty much any job that has a dollar amount more than you are willing to toss in the toilet should have some kind of contract attached at least stating that amount, the date when client deliverables are due and the date on which you deliver the finished work, and shows some kind of agreement between you and the client on the terms.
Getting payment, whether up front, in thirds or halves is only a small part of the contract and the contract itself does not guarantee payment. For that, as a business, you should know how to investigate your clients and determine the risk of non-payment. For instance, maybe you have all new customers pay at least half up front, depending on the size of the job. If it is a larger job it is done in installments as each section of the job reaches a milestone as described in the contract.
Larger jobs, like government contracts or very large multi-discipline rollouts, require you as a designer to prove work has been done.You have to submit payment applications with deliverables in some form in order to get your money for that segment of work.
For some jobs, where materials are required (such as a rush trade-show-booth-in-a-bag) the initial investment the customer makes is for the hardware required so you aren’t laying out your own money for that stuff.
There is no set in stone contract. If you are starting a freelancing business it may behoove you to talk to an accountant and see what they suggest regarding not only contracts, but all fees, taxes, and other things you have to know in order to be a business owner. Most importantly, how to separate your personal assets from your business assets in the event you make a bankrupting-scale mistake. Most states in the US also have a Small Business Administration that offers all kinds of information and seminars about running a small business in your state.
As far as Purchase orders go, these are, in effect, contracts. But since they usually only consist of a PO number and maybe a rudimentary description of the item being purchased, and since they originate from your client, you have no control over the terms. You don’t know from looking at a PO if the client pays out weekly, biweekly, monthly or quarterly. That can make budgeting a little tough. They also don’t always specify a due date and they almost never spell out the client obligations in order to meet that due date if there is one.