The main thing about any eclipse is to get out and see it, so here's some information on the best ways to do that. Particularly in a brief total solar eclipse, you may well decide to just relax and enjoy the event, leaving photography to the experts; but if you'd like to have a go, there's some information on eclipse photography here too.
Careless observing of a solar eclipse can cause permanent eye damage; so be sure to read about eclipse eye safety.
If you're going to see a solar eclipse, the first thing to do is find a good place to see it from. For a partial solar eclipse, which is visible over a large part of the Earth, this should be easy; but more care is required if you want to get the best from a total, annular or hybrid solar eclipse.
You will see from the maps of total solar eclipses that the path of totality is quite narrow; if you're not in this zone, you'll miss the total eclipse entirely, and only see a much less interesting partial eclipse. You want to be fairly close to the centre of this path to see a decently long eclipse — you don't have to be right on the centreline, as the duration of the eclipse falls off quite slowly at first as you move away from the centre; closer to the edge, however, it drops much more rapidly. As a rule of thumb, if you are in the middle third of the path you will see a total eclipse within about 90% of the duration on the centreline.
Of course, a clear sky is essential, so try to get to a place with the least chance of cloud. Study the weather reports, and try to find a place with the best chance of clear skies.
Finally, and this may sound obvious, pick a place with a good view of the sky where the Sun will be during the eclipse. Go there the previous day at the same time to see where the Sun will be, or work it out with a compass. If you're going to set up a solar projector (see below), a practice viewing the day before will help you to figure out the best setup. (And you might see Sunspots!)
One tip — stay away from streetlights! Automatic streetlights may come on right at the moment of totality, which will spoil everything if you're under them.
So, once you're in place, what's the best way to view the eclipse safely?
To look at the Sun directly, you need eye protection; otherwise you risk permanent eye damage. Sunglasses are nowhere near good enough; properly designed solar filters (such as eclipse viewing glasses), made and certified to appropriate national safety standards, are needed to view the Sun in any partial or annular eclipse, and the partial phases of a total eclipse. See the eclipse eye safety page for more information.
Ad-hoc methods such as welder's goggles are not safe unless you know exactly what you're doing; you're probably better off just getting purpose-designed eclipse glasses. But with fake goods of all kinds flooding the market these days, make sure you get your eclipse glasses from a reputable source.
The total phase of a solar eclipse — when the Sun is completely covered by the Moon — can and should be viewed without any filters. The naked eye view of totality is safe and is the most awe-inspiring astronomical phenomenon you are likely to see. Just remember to look away and put your eclipse glasses back on as soon as the Sun returns.
The most recommended safe methods for viewing the Sun have one thing in common: they involve projecting the Sun's light onto some surface, so that the Sun is viewed indirectly. Be careful when setting these types of experiment up; the temptation to look through the setup to get it lined up must be resisted! The golden rule is that your back should be towards the Sun at all times.
The simplest method is a pinhole camera. Make a small hole in a sheet of card, and hold it about a metre from another sheet of card, with your back to the Sun; the Sun's light will be "projected" through the pinhole and form an inverted image on the second sheet of card, which you can then look at safely. Don't look through the pinhole! You can adjust the size of the image by adjusting the separation of the cards.
Unfortunately, the image projected will be very small; just over 9mm for every metre of distance between the pinhole and the screen. You can improve on this by putting the screen a long way back from the pinhole, but you'll need to have lots of shade around the screen for the faint image to be visible. One idea is to put the pinhole in the window of a dark room, surrounded by curtains, or to use a tent. Making the pinhole bigger won't help — it will just make the image blurrier. A small pinhole will give you a sharper image, albeit dimmer.
A better technique is to project an image of the Sun using a telescope, or one side of a pair of binoculars. Place a sheet of cardboard around the objective (the big end) of the telescope or binoculars, to act as a shade for a second sheet positioned behind the eyepiece, about a foot or more away. The danger of this method is the temptation to peek through the telescope — you must not do this, even for a moment, or you will risk serious permanent eye damage!
Setting this up can be a little fiddly, especially as the Sun is moving. In fact, you will discover that the Sun actually moves quite quickly across the sky — it will move its own diameter in about 2 minutes. Having a telescope or binoculars on a tripod, with a sunshade and screen directly attached, should help. The whole thing should be adjusted by trial and error; you can adjust the focus by means of the focusing knob, and by moving the screen. Bear in mind that the projected image will be inverted — i.e. upside-down and left-to-right.
The projected image may be too bright for comfortable viewing or photography; if so, you can dim it with some tinted plastic, or an ND filter, in front of the telescope; or just block out part of the lens with a piece of card with a hole in it, or even just a square of tape as shown here.
Remember that you can experiment with the projection technique on any sunny day — and this is a good way to look for sunspots! In fact, you should certainly build and try out your Sun projector in advance of an eclipse, to fine-tune it.
The projector shown here was used to capture the image, shown above, of the partial Solar Eclipse of 23 Oct, 2014, and the huge sunspot region AR 2192.
If you are under trees when the eclipse is happening, you'll have a good chance to see images of the crescent Sun on the ground beneath them. This is because the gaps between the leaves act like pinhole cameras to project images of the Sun on to the ground. If the trees are tall enough, these images can be quite large.
Finally, the Sun during totality is a beautiful and spectacular sight; but don't be caught out by the end of the total period! Find out how long the total eclipse is going to be where you are, bearing in mind that a difference of a few miles can make a huge difference, and be prepared to look away and put solar viewing glasses back on at the first signs of the returning Sun.
Viewing lunar eclipses is easy and completely safe. The Sun is safely below the horizon; above you in the night sky hangs the Full Moon, slowly being stained a dark red colour by the encroaching shadow of the Earth... Gaze and enjoy! (In case you're wondering, the Full Moon is about 400,000 times dimmer than the Sun.)
It's worth thinking about your location. A lunar eclipse is visible over half the world, so as long as you're in that half, you should see it; but make sure you have a clear view of where the Moon will be during the eclipse. If it happens in the evening, just after sunset, the Moon will be rising in the east; if it happens just before dawn, the Moon will be setting in the west. If it happens in the middle of the night, the Moon will be high in the sky, so the view shouldn't be a problem.
As with any eclipse, the main thing you need is a clear sky, so if possible look for a location with a lower chance of clouds.
Any night-time astronomy is greatly enhanced if you can get away from all the light pollution and smog in towns, so getting out into the country is a good idea; but having said that, a lunar eclipse should be easy to see from any back garden. With the eclipse being visible from the entire night side of the Earth, all you need is a clear sky.
No special equipment is needed; a lunar eclipse is great for naked-eye viewing. Dress for the night temperatures and take a flask of something warm. The biggest danger is tripping over something in the dark, so watch your step!
Binoculars can help you to see the full beauty of the Moon, and can reveal a fascinating view of the lunar craters — try looking at craters around the edge of the Full Moon. Though, in fact, looking at the Moon when it's half full is even better; the craters stand up in dramatic relief with the Sunlight slanting across the lunar landscape.
Eclipse photography is a slightly tricky business, at least until someone has figured it out for you — since other people have written about this far better than I can, I'll just point you to some resources.
The links below provide useful information on photographing and videoing an eclipse, including shutter speeds, filters, and so on.