During recent travels, I had the opportunity to take many photographs, and also to see many other people taking photographs. While I am not an expert travel photographer, I noticed a few things worth commenting on.
For most people, the best type of camera for travel photography will probably be mobile telephones (assuming that they are recent models with good built-in cameras). These will strike a good balance between image quality and portability—’the best camera is the one that you have with you’ (i.e., you cannot even take a photograph if you do not have a camera with you). If your mobile telephone is connected to GPS services, your photographs will also be geotagged, so you will have a record of where you physically took each picture.
If you want to take photographs in low light, such as at sunrise or sunset, you could still use a mobile telephone, but be sure to stabilise the camera as best possible. Setting your telephone on a solid, stationary object (e.g., a low wall nearby), or else leaning against such an object while handholding your telephone, are two easy ways to do this. A more expensive but also more flexible option is to use a separate, dedicated camera that has image stabilisation built-in; most recent point-and-shoot models should have this functionality. This will allow you to shoot from anywhere, without needing to find an object against which to stabilise yourself.
For very low light scenes, such as cityscapes at night, you will really need a solid, stationary object or a tripod. The exposure durations will be too long for any camera- or lens-based image stabilisation to be effective. Using a flash will not help in these situations; a flash gives only very limited lighting capacity. (You cannot point your flash at a subject that is, say, 50 m away and expect to get a good photograph. The flash simply will not have enough power to light up the subject at that distance, and you will end up with a dark, unusable picture.)
If you decide that you want to take photographs with better image quality than you can get from your mobile telephone, look to either a good point-and-shoot camera or a small DSLR. If you can afford to carry a dedicated camera with you, the DSLR will give better image quality because it will have a larger sensor (and better supporting electronics) than you will find in a point-and-shoot.
If you are new to photography, the best investment you can make is to spend some time investigating and then practising photographic skills. I mentioned stabilisation above, but composition, exposure, and depth-of-field will also be important aspects of photography to become familiar with. Any photographic equipment will have some level of capability, but will do little good if the photographer does not have at least some basic skills.