You're very lucky that you get to pick up your bun without a problem. Buns should have their nails clipped at least every month so I don't know how long you it has been but since you said it's long then it must have been a long time. Anyways, I wouldn't use human nail clippers because their designed for humans and not pets. I would go for cat clippers or search rabbit clippers on amazon because those work the best.
It sounds like this is the first time you're nail clipping so I will guide you through it. There are two main types of nails, clear and dark nails. Clear nails are the easiest because you can see the quick the easiest. The quick is a vein which should NOT be cut at all, if cute the rabbit can bleed and it will be painful. If it does start to bleed you can use household products such as flour to stop the bleeding and if it continues to bleed go straight to the vets. When you're cutting cut at least 1cm away from the quick and if you're still unsure only cut the part of the nail that looks like a hook and is far away from the quick to be safe.
If the nails happen to be dark, then you will need to do the pressure technique. When cutting the nail make sure not to go to far up only cut of bit by bit, put light pressure with the clipper on the nail, if they move then you need to move down a bit, do this again and if they don't move put a bit more pressure on it (if they move repeat above), then you finally clip. Dark nails are the hardest because you have to guess where to cut really, so if you're unsure about it go to the vets to let them do it instead.
My bun gets really annoyed when I clip his nails so try to calm them before doing it (since it's his first time) and for every nail cut give them a treat (I give mine dried papaya every time I cut a nail) so it teaches them that if they co-operate with you they can get more treats.
Hope this helps :goodluck