when do you start using/ paying a solicitor?

[MrsPuddle]

Do you wait until either side makes an offer, or before?

I would engage one as soon as possible, the estate agent will usually ask who you are using.

Once you have made an offer to buy a house and the seller accepts your offer. You need to hire solicitor who deals with property (conveyancer). You'll need to pay for disbursements up front and a small percentage. They will send you a pack and you sign a contract with the solicitor.

We did research and obtained quotes for our chosen solicitor as soon as we put our house on the market.

We've had some terrible conveyancing solicitors in the past and they really do make or break the experience

Do your research. As pp says, a decent conveyancing solicitor makes all the difference. Ideally you need a practice with enough staff to do the background grunt work and chasing.

When we make an offer on a house the EA asks who is your solicitor? That prompts us to phone around to get quotes and engage a solicitor within a day or two.

You don’t normally need to pay much upfront to the solicitor. Most of the fees become payable at Exchange of contracts which is also when you generally pay the EA fees (unless Purple Bricks or one of those where you have paid fees upfront).