Those bills come to just under £3k with scope to cut down (I'm of the opinion that, even if you can easily afford it, it's always worth reducing the cost of boring essential bills unless you literally have more money than you know what to do with as it leaves more to save for the future and for nice things/fun stuff).
So you have around £1600 spare plus your partner's earnings, with grocery shopping as the only essential major omission.
So unless you're already saving £1k+ each month, it's likely that your non essential spending is significant and also needs looking at. Or is the £4600 your total earnings?
In any case, Moneysavingexpert.com has a good budget planner, look for the money makeover which walks you through reviewing all your expenses.
Once you've identified what you 'should' be able to save, set up standing orders to move it to separate accounts when you get paid, in case you're at risk of frittering apparently spare money in your current account.
Split your savings into at least two pots. One to cover fluctuations in monthly spending, eg insurance, Christmas, car repairs, white goods replacement, so likely to get spent in the short to medium term and a separate one for the long term savings you want to build up.
Perhaps also have pots for things like car replacement and holidays as these are expensive and have a discretionary element to the amount so a good way of ring fencing what you can afford to spend on these items.