First of all, congratulations on your new house and garden! It sounds lovely!
The first rule of gardening is that fighting weeds is a constant battle between about April and October! Don't assume that because you've cleared the blighters once, you can turn your back. You need to take a Mad Eye Moody stance on this - CONSTANT VIGILENCE! Once you've cleared an area, make sure you hoe it every week to stop anything growing back. Even if the weeds are only tiny, prevention is better than pulling out gigantic brambles. Things like horsetail are a battle that will take some time to win - you need to hit them, hit them and hit them again and eventually you will prevail. But they are a notoriously tough enemy.
The good news, however, is that once you have cleared, doing this little and often type of job doesn't take very long at all! Plus, if you are really busy, then you can then get help from mulch. Putting a thick layer of brown goodness all over the beds really does amazing work in reducing the need to weed and in helping the plants and the soil.
With the veg patches - if they are empty (apart from weeds), you could think about using black weed sheeting to cover them. This will kill the weeds with the lack of light, and then you can dig over or mulch in the autumn (and maybe even get some overwintering crops in). If your DC are old enough, you could even give them one each as their little 'patch' to look after.
Finally, this type of gardening - clearing, digging- is really, really hard work. Little and often is better than big stints, which are exhausting and a bit demoralising. If you can spend an hour out there with your DH and DC three times a week, then you'll be surprised how much you can get done!