Before you provide your resignation to your current employer, make sure the employment agreement with your new employer is signed and countersigned, and that both parties have a copy. This means you should have a copy of a document (physical, PDF, etc.) that has the entire employment agreement, with your signature and the hiring manager's signature (and any other signature fields filled in, such as a company officer's signature, etc.). Make sure no signature fields are blank.
Next, inform your point of contact (i.e. your HR contact), via e-mail (i.e. so a written record exists), that you will tender your resignation with your current employer and commence the notice period once you receive formal confirmation that:
- Any/all outstanding conditions (even those not mentioned in your contract, such as good references, criminal background record check, etc.) have been lifted.
- That the signed + countersigned contract is now 100% unconditional.
If your future employer states that they can't provide the unconditional offer until they've completed the reference check a few days/weeks/months from now, that's fine: you can wait and continue working with your current employer until they're ready, and you can serve your 2 weeks notice or equivalent when you've received an unconditional offer, in writing. If they insist on you gambling with your livelihood and having to accept a conditional offer, I'd personally decline (though your needs/situation may be wildly different; I err on the side of staying employed and having an income during this bear/garbage market period). If they push you based on the terms of the contract (i.e. "you said you'd start on XYZ date"), you can reply with "that was under the terms explicitly written in the contract: no conditions; but conditions have been added as per previous contact/e-mail/phone-calls/etc.; so we'll need to wrap those up first).
I've had an employer try to pressure me to resign early on the references and criminal record check, and I just responded with "while I'm confident my references will be positive, I can't control them" and "background checks could be off if someone stole my identity, for example; it is indeed possible to cheat an honest man". Never tender your resignation until you have formal written records of unconditional offers of employment, as they're all you'll have if you have to sue your employer for breach of contract or illegal/unethical hiring practises if they don't hire you after all, for any reason (i.e. didn't like your references, you popped up saying something "offensive" on Facebook, they don't like your political affiliations, they ran out of money and are trying to back-out of the job offer, they don't like your face, etc.).
"Think like a monkey: never let go of a branch in your left hand until you have another firmly in your right hand".