Golden Orb Spider . . .
NOTE: The original image above showed the spider facing down but it didn't appear as fearsome as it does this way up.

It's during the cold, wintry time of year, that we tend to want to stay in bed that little bit longer, unwilling to brave the cold, frosty air. And rightly so. That is, of course, unless you're keen enough to get out of bed nice and early in order to get the best photos. Then you'll stop at nothing, right? 
Anyway, winter has got to be the best time of year for photographing spider webs. Especially if it gets a bit foggy or frosty, or at least if there has been a heavy overnight dew. It's also at this time of year that the sun rises a little later, so you don't have to get out of bed as early as you do in summer and it takes a little longer for things to thaw out, in the cool, damp, frosty air and there is also more chance of the air being quiet and still. Another good point is that during the colder months the sun isn't quite as high in the sky, providing a more favourable light source.
Spider webs come in all shapes and sizes and generally, the spider will stay in the same area for a few days, if it gets a regular enough meal. So, if you see one when you're out and about, then return the next morning, chances are it will still be there and hopefully in reasonably good condition. 
You don't need any special equipment for photographing spider webs, but it would be ideal to be able to fill the frame with it. So, to be able to get quite close, would surely be an asset and if you have a lens that will enable you to get even closer, that would be better still. 
The best time of the morning is just as the sun rises. You should have up to a good hour after that to get the best light. Once you have found the spiderweb you want to photograph, you must then look for the position and angle where you can get the best light for your subject and that's when the entire mesh of the web is suddenly lit up with sparkling little beads, that shine like a necklace of tiny little gems.
A dark, out of focus, background is ideal for this, as it will make your spiderweb come alive even more, but try if you can (if you are setting a large aperture), to keep your camera on a parallel plane with your subject, or you could send one end of it out of focus and your shot may not turn out the way you planned.
If you are shooting in the direction of the light source, be aware of the possibility of lens flare. If you can get really close, like 'Macro' close, try photographing single strands of beads. You will be surprised just how magical they look. 
As I said earlier, it is a bonus if you find a web in good condition, without a lot of broken strands and even better if the spider is sitting on the web. Golden Orb spiders usually sit in the centre of their webs, waiting for a victim, whilst others will find a curled up leaf or choose some other place to hide during the day, till they again venture out at nightfall.
They may be sometimes fearful but they do make a for very fascinating study.
Like most other subjects, it is the amount of effort you put in that determines the result, so don't rush things. Take as many shots as you can, from various viewpoints and angles. If your camera allows for manual operation, switch to aperture priority and select different aperture settings. As a general rule of thumb, I would advise that you work on settings from about f2.8, f4 or f5.6 and as always, planning is very important and the more planning you put in, the better the result. But above all, enjoy it!

For your information, the two spiders I have featured here are the Golden Orb Spider. They were named the Golden Orb Spider because the web they spin is actually golden in colour. The first image (top), shows an immature spider. It is difficult, in a photo, to tell that it is a juvenile because they resemble direct clones of their parents. In this view is about 1 to 1 1/2:1 magnification. In other words, it's about one and a half times its normal size. The little one (above), is about the same magnification.
Some spiders are nocturnal, mainly feeding at night then hide during the day, but these guys remain on their webs and continue to feed throughout the day. When they are not feeding, to relieve the boredom, they quietly go about repairing any holes in their webs and are not too concerned about the odd human who passes by and stops for a look.
They can inflict quite a nasty bite, but I haven't as yet heard of anyone being bitten by one.