First off resize as you are doing in Image/Image Size. Then, using File/Save for Web & Devices and the dialogue box will open. Select the Optimise Tab. I have captured the top right part of the box
You can use the settings as shown. You will see a tiny arrow to the right of Preset, click that and you will have the option to select the optimum file size. A second option is to set your new Image Size and then File/Save As. A dialogue box will open allowing you to set the file size (roughly). If Saving for Web the image will lose it's metadata but Save As will retain it. HTH