I would like to take pictures on my 35mm film camera that are wider/longer. Is there any way to do that on a film camera without just cropping the photo? Can I use different sizes on film on my camera or would i have to buy a whole new camera?
The size of the photo depend of frame, which is inside the camera, usually between shutter and film. So to make panorama like photo (if this is what you mean by "wider/longer") you should crop the photo. Here is example of this "frame"
Or buy new camera which is created for such kind of photos.
EDIT: You can use one "hack" putting two bands of nontransparent material on the upper and lower part of this frame. But this will move the film slightly behind (loose focus), put in dangerous to scratch the film. And at the end there will be no different than crop in postproduction :)
Edit 2: "Using" answer from Horitsu you can buy anamorphic lens. Unfortunately those lens are quite expensive (> 500 USD/Euro) and often they come as adapter to particular lens. The downside is you should use anamorphic lens with same ratio when you print your photos on paper or stretch the scan in postproduction.
A roll of film is fully "photosensitive". It's not the case that there are predefined "frames" where a photo can appear – the size of the frame that is exposed depends explicitly on the camera – you can't buy "panoramic film". There are cameras that expose different frame sizes, but the standard size for cameras using 135 film is 24mm × 36mm.
If you are looking for a camera that exposes a wider image, take a look at the Hasselblad XPan or XPan II (aka Fuji TX-1 and TX-2 respectively). This is a discontinued camera, and a used (working) one will not be cheap, but it exposes much wider frames (24mm × 65mm) by essentially using medium format lenses (or rather, lenses that project an image circle wide enough for medium format). This is a manual focus, rangefinder camera, and will be a very strange beast for someone new to this kind of camera.
The Lomographic Society offer some cameras that will expose the image over the film's sprocket holes. Look for their Sprocket Rocket camera series – these are much less expensive cameras, and could provide for some fun experimentation.
There are also rotating-lens panoramic cameras, where the lens is in a kind of drum that rotates when you take a photo. I own one of these – a Noblex 135 S – and I love the results I get with it. The Lomographic Society also offer one or two models of this type – check out their Horizon camera series. This is a quirky style of photography, with its own idiosyncrasies – and not to everyone's taste.
Note that developing film is the same regardless of what camera is used, but printing/scanning images may not be as straightforward when the frames are not the standard 24mm x 36mm.
Take a look at these related questions also:
Buy a whole new camera if want to do it in-camera. 35mm panoramic cameras are special-purpose and built for this one task.
I could shoot 3:2 (C), 16:9 (H), or 3:1 (P) on the same roll with my old APS Canon Elph, but that was simply digitally recording on the APS-C magstrip how you wanted the frame to be cropped for prints. And of course, these days, APS is discontinued.
It's a bit arts'n'craft-y but you could just shoot multiple frames and overlap the prints in an album, which is what I used to do back in the day. Or you could go Cubist and do a Hockney joiner. :)
These days, you could also get the roll scanned and then panostitch the frames together with software.
There are purpose-made cameras that shoot very wide format shots on 35mm film, e.g. the Hasselblad XPAN, which shoots a 24x65mm image (standard 35mm format is 24x36mm).
Another option is a medium format camera with a wide-format 35mm back, such as the Bronica SQ series with the 135W back. It shoots about 24x58mm (I'm estimating the second measurement, but it'll be just shy of 60mm since the system is designed for 6x6cm rollfilm). These backs seem rare and quite expensive, which is unfortunate, since I'd love to have one for my SQ system. :)
Of course, you can just shoot standard 35mm images and crop the top and bottom (some cameras will have a built-in mask system that will do this for you, but you can simply do this yourself). You can also shoot larger formats, like 120 rollfilm, and do the cropping yourself and maintain higher quality, since you're using a larger negative.