Binoculars' low light performance isn't entirely governed by the size of the objective - front - lenses, it's largely determined by the exit pupil. You can work that out very easily by dividing the diameter of the objective lenses by the magnification, so 10 x 50 = 5, 8 x 32 = 4 etc. Our own pupils can expand to about 7mm in poor light, and 'night glasses' are usually 7 x 50 or 8 x 56 to take advantage of this, but these binoculars are fairly specialised and are big and heavy. The quality of the optics and coatings also impacts light transmission, and it can be fairly significant.
10 x 50 (if you can hold them steady, and are comfortable with the size/weight) and 8 x 40 both have an exit pupil of 5, and are good choices for most purposes. 8 x 30 and 8 x 32 are also very good, and a bit smaller/lighter. There are other alternatives, but these are probably the most popular and are available to suit just about any budget.
The only binoculars I'd caution against are the very small (usually 8 x 20, 8 x 25 and 10 x 25) 'pocket' models. They have a place, and are very convenient to carry, but trade off low light performance and can be difficult to hold steady. I do have a pair of Leica 10 x 25s, but I bought them in Africa - where the light is usually strong - and accepted their limitations because I wanted very small/light binoculars for backpacking and walking in the mountains/bush. They're stunning, but very expensive, and I do have a larger pair of 8 x 42s too.
Binoculars have to 'fit' you comfortably, so I'd go to a shop with a wide range and try as many as possible outside, not indoors. You'll find something to suit you, and you shouldn't have to pay more than £100 tops for a decent pair. Some of the lower priced models might be fine too. The high end Leica, Zeiss and Swarovski binoculars are fantastic, but they're designed for people who want the ultimate in optical and build quality for heavy use, or who just have a lot of money. Bit like cameras!