An important distinction is between historic and contemporary graveyards. More recent graveyards are often arranged with roads within them wide enough for a motorised hearse, more accessible paths (e.g paved or gravel, etc.) and the monuments are usually in better condition (but don't climb on them), however they are also in active use, so people will be visiting them as mourners visiting loved ones, and it is even more important to remain respectful of other cemetery-goers, and not to do anything that could impact on its use. Historic cemeteries often have very interesting and sometimes quite large and elaborate statues, mausoleums, tombs, etc. but they also tend to have less accessible paths, and the monuments can be in a state of disrepair; I know several locally where some of the mausoleums are in such a dangerous state of dereliction that they have to be fenced off with warning signs, and others where some of the graves have sunken downwards - in such places, keep to marked paths if possible, and avoid entering the mausoleums, especially if they look unstable or are closed off. I know they're enchantingly gorgeous, but that's not worth ending up as a permanent resident...
There is also a difference between municipal or council-run cemeteries and ones attached to a place of worship. Obviously, if you are in a graveyard associated to a church, cathedral or chapel, one should be respectful to the place as a religious place as well as a place of rest for the dead. The church may well still be in use, even in historic graveyards with no new graves and is important to both be respectful of those attending the church and not to do anything that might disturb them; do not be noisy, for example, especially when there is a service of any sort in session, and remember that services are not only on Sunday mornings!
So here are my guidelines to cemetery behaviour. This is based around my experience in the UK, and other cultures have different etiquette for visiting graveyards.
1) Do not clamber on the statues/grave-stones/grave-markers/tombs. From a practical standpoint, you could damage them. Yes, a lot of them are made of stone, but stone weathers with age, and not all stones have the same sort of strengths. A lot of times it is the details of carvings which become fragile, and some stones become soft, friable or flaky with weathering. Acidic rain from the industrial revolution onwards has had a very depressing impact on specific kinds of stone, especially.
From a perspective of being respectful, these are people's burial places and it can be considered disrespectful to those interred and their families to be using their markers as props for photo-shoots, something to clamber on, etc.
2) Do not drop litter. If you are having a picnic, or bringing any kind of food or something with a wrapper (even if it's just the plastic over a sketchbook, for example), either dispose of it in a bin, or take it home with you. A lot of cemeteries and graveyards have bins provided, especially ones which get frequent traffic, and ones still in use, but even if they don't, that is no excuse to be slobbish and leave litter.
3) Don't let your dogs foul the graveyard, and if there's a sign saying no dogs, then respect it. Personally, I wouldn't bring a dog into a graveyard at all, and if I did, I would keep it on a lead, especially if its liable to go chasing the squirrels or something, to preserve both the peace of the place and the statuary and tributes from getting knocked or damaged. If you do bring your dog into the graveyard, and it uses it as a toilet, please clean up after it. Just imagine the person who has to use a strimmer on the grass finding concealed dog faeces.
4) Respect the peace of the graveyard as resting place. You do not have to keep to absolute silence, but using quiet voices and not being raucous or to bouncy is probably a good idea, especially in one where people have recently been interred, and where people might be visiting as mourners. Treat it as a garden of quiet contemplation, not a public playground.
4) Don't let children play in the graveyard. Some children can be trusted to be well-behaved and quiet within graveyards, others can't. Don't let children climb on the stones, run around very excitedly, or otherwise behave in a manner that might damage the graveyard, cause injury to themselves (recently a boy was crushed to death by a gravestone as can be read about ::here::). Graveyards are not a safe place for play, especially as tombs can become unstable over time.
5) Leave tributes alone - don't mess with anything anyone has put on a grave. Absolutely NEVER take anything left by mourners on a grave. The only exception I would see is if a real candle was lit and something had fallen or was in a position where it might cause a fire-hazard.
6) Don't use it as a place to host your super-spooky 'ritual' or seance or whatever. Most graveyards are associated to a church, and it is disrespectful to them as hallowed ground places of Christian worship. It is also not a good idea to do this in municipal/council-run cemeteries, as many people would consider it disrespectful. You can do a seance in your own home. Sometimes ghost-hunting groups can get permission to engage in their practices with permission from whoever runs the cemetery, but do not do anything of that nature without permission.
7) Pay close attention to the opening and closing times. Many graveyards and cemeteries shut at night due to problems with drunks and delinquents being a nuisance after dark, and if you stay too late, you run the risk of both being locked in, and being considered a miscreant. Don't try and jump the fence after closing; respect that whoever runs it is entitled to set their own opening hours.
These are the 7 things I would give as 'rules', but also check to see if there are signs by the entrances specifying additional rules. Just because I haven't mentioned something, that does not automatically make it a good idea, and if in doubt, it's better to be safe than sorry.
Notes for photographers
I would avoid are taking photographs of the text on markers; to me, that is the private details of whomever is buried there, and is for their family, not for everyone to gawp over, but that is my personal preference. As you may note from my photography, I tend to either photograph only a small detail, or the whole cemetery, rather than focusing on specific stones.
I also would never pose, or act (in the theatrical sort of way) as a 'widow' or 'mourner' at someone's specific gravestone; that person probably already had real mourners, and it seems distasteful to play at being mourner when someone probably suffered real grief and pain over the person that was buried there. I would not encourage anyone modelling for me to do so either.
In a similar manner, I would not encourage anyone to model, nor model myself, in an overtly sexual way. I think this would be disrespectful to those interred, and to those visiting, especially those who are going there for a sombre purpose. Mix the iconography of sex and death, by all means, but don't be disrespectful in a cemetery to do so.