Fill in a P46 and hand it to your new employer. Your old employer should hand you 2 parts of the P45, the 3rd part, intended for the tax office, is no longer needed as the old employer should inform HMRC electronically of the same details.
Your new employer should inform HMRC electronically that you have started with them and supplied a P46. You will probably be put on basic rate tax until HMRC then inform your new employer of your tax coding.
You will need to retain your copies of the P45, which together with the P60 you should get at the end of the tax year from your new employer, will show your earnings and tax paid, to enable you to make any claim for a refund of the overpaid tax.
If you want your tax to be correct from when you start with your new employer you must give them your P45 instead.