Do some legwork ahead of time
You'll want to do some serious studying of your local market. Make sure that there are properties available that you'd like to move into, and get a sense of how much they cost. Get your mortgage pre-approval, and have your financial documents in order so you can move quickly.
In the meantime, work with a real estate agent to prepare your own home for listing. You'll want to be as realistic as possible in valuing your property, both to ensure it sells quickly and to get a sense of your potential proceeds/shortfall.
Use one realtor for both transactions
Buying and selling real estate is complicated, and when you're doing multiple deals at once, there are lots of moving parts that can throw your schedule off track and threaten the deal. If it's possible, it makes sense to work with one real estate agent for both transactions, since they'll be incentivized to get both deals done on schedule.
Bonus: Some realtors will give you a discount on their sales commission if you use them to help you buy a home, too.
Be cautious about waiving your contingency
The Ondillas' experience illustrates how tough it can be to find sellers willing to accept a contingency clause, making the purchase conditional on the sale of your home. If you're relying on the equity from the first home to finance the purchase of the second property, however, waiving the contingency comes with risks. Without a contingency clause, if the sale of your first home falls through and you're unable to purchase the second property, you could lose your earnest money deposit.
To keep your contingency, you may have to increase your offer price or find another way to sweeten the deal. For example, the Ondillas agreed that if they couldn't complete the purchase, they'd give the sellers $2,000 to put toward mortgage payments while looking for another buyer.