Tiff and PSD are lossless...so you won't loose any quality when you save a file in these formats. If you save a file as JPG, you are compressing it and loosing some of the information. With Photoshop, you can choose the level of JPG compression.
Even when you set your camera to save JPG files, you are losing some of the info.
Once you have the file into photoshop, you should save it as PSD to preserve the quality. When you want to send it for printing, save a copy of it in JPG format but don't overwrite your working copy. Then when you want to upload it for web viewing, use the save for web option and adjust both the image size and JPG compression.
If you plan to do a lot of shooting, you should come up with a system and stick to it. For example, you could save all the files right out of the camera and keep these as your "negatives". Then save a working copy to play with in PS. Then save a copy for printing and a different copy for web viewing. This will take up your memory space rather quickly so you might want to think about archiving your files onto CD or DVD etc.
You could go with a more basic work flow if that suits you better but try to think about it rather than just filling up your computer with random files...it will make it easier for you in the future.