Almost every move-up buyer faces this, and there is no universally right answer. There is a right answer for your risk tolerance and your financing, and those two pull in opposite directions.
The two orders
Buy first, then sell
You secure the home you want and sell without a deadline hanging over you. The risk is financial: if your sale comes in below expectation, or does not sell, you are carrying two properties. This is the order that ends badly when a market turns.
Sell first, then buy
You know exactly what you have to spend and you negotiate from strength. The risk lands at the other end: a firm closing date and nowhere confirmed to go. In a fast market that means compromising on the purchase, or a rental and two moves.
In a seller's market with low inventory, buying first is the greater risk, because the thing you cannot control is finding a home. In a buyer's market with rising days-on-market, selling first is the greater risk, because the thing you cannot control is your own sale.
Read the pocket you are actually in rather than a general rule. See Observing the Market.
Bridge financing
A short-term loan covering the gap when your purchase closes before your sale does. It lets you close on the new home using equity from a house you have not been paid for yet.
It requires a firm sale. Lenders bridge against an unconditional agreement of purchase and sale, not against a listing or a hope. This is the single most important thing to understand about it, and it is why buying first without having sold is a genuinely different proposition from buying first with a firm deal in hand.
It is short, typically up to 90 or 120 days, arranged with the lender funding your new mortgage.
It is not cheap, and it is not ruinous either. Expect a rate meaningfully above prime plus a setup fee, on a balance you carry for weeks rather than years.
Conditional on the sale of your property
The safest structure available to you and the weakest offer you can write. In competition it is close to unusable, and where a seller does accept one, expect an escape clause letting them keep marketing while giving you a short window to firm up if another offer arrives.
Worth knowing what that feels like from the other side: the seller has not really stopped selling, and you may get a phone call giving you forty-eight hours to commit unconditionally. Decide in advance what you would do if that call came.
Matching the dates
The cleanest outcome is closing your sale and your purchase on the same day, or a few days apart. It takes coordination between two sets of lawyers and two lenders, and it removes the need to bridge at all. Where it is possible it is worth designing both transactions around.
Before deciding the order, ask your lender two questions: will you bridge for me, and on what conditions. The answer frequently decides the sequence for you, and it is a five-minute call that saves a great deal of speculation.
Selling and Buying at the Same Time covers this from the other direction, including what your offer looks like to the person receiving it.