Unless the client specifically asks for something else (and knows what they're doing), then just give them JPEG.
If they are going to be further editing the photo (they would need your permission or transfer of rights), then they may benefit from a TIFF file (or even PSD). But if they are just going to print it, or display on the internet etc, then jpeg is probably OK.
Keep in mind that there are other factors that affect the size of the file. The file type obviously, but that is mainly because JPEG files are designed to compress images. When you save as a JPEG, it strips the unneeded data and compresses the image. This lowers the quality, but usually not so much as to be visible when viewing the image or print.
Further to that, you can set the compression/quality levels. When you export, there is a slider for it. The higher the quality, the less compression and thus the larger the file. The lower the quality (higher compression) the smaller the file size.
But there is also the image size that will affect the size of the file. For example, if you save the image at 5760 x 3840 pixels...then it will be a much bigger file size than if you save it at 1800 x 1200 pixels.
This is where you really need to consider the uses for the image. If someone is going to print it, how large will they need to print it? The more pixels, the easier/better it will be to make larger prints. But if they are just going to view them or post them to the internet (Facebook etc.), then 900 pixels wide is probably large enough.