To get these sort of shots you are going to need a camera that will allow you to have some degree of control. Auto modes are no good for this. However, it is not that easy as you will need to have some knowledge of how to get the shots too. Essentially you need long exposure times and because of this you will also need a tripod or use a wall or fence etc to support the camera.
Three types of camera to consider.
1 dSLR. Very capable, hugely adaptable with interchangeable lenses and can be very costly. Entry level models are not too dear and will do the job. All of the companies will make decant models - go for the one that is easiest to access and change the settings and feels best when you handle it.
2 Advanced compact cameras. Will offer the control you need in a compact body. Cost much the same as entry level dSLR's. Cameras to consider are the Canon G11 (earlier G10/G9 etc will do the job), Ricoh CX2 or CX3, Panasonic LX3, top end Sony and there are others too.
3 'Bridge' cameras - look like dSLR's, but have fixed lenses with huge zoom ranges. Similar prices to entry level dSLR/advanced compact. Same principals apply as for dSLR's. Fujifilm, Panasonic FZ series, Nikon and Olympus amongst others.
Big savings to be had by buying used kit.
You might be able to get these shots with an 'ordinary' camera, but it will be hit & miss and depend a lot on the location and what lighting is like.
ive just bought a sony H20 and it has all the manual settings not a dslr though. i take pics like that. you need a bit of skill and tripod to do them though. its a great camera not too big and looks the part. good for quick switch on and snap pics and also like above...£179