I recently purchased an Olympus Trip AF 35mm point & shoot. I have tried multiple different types of film and have brought the rolls to my local camera store to get developed. Every time, the photos come out with numbers in the bottom right corner. Is there a way to prevent these numbers from being there when I have the roll developed? See example below.
The numbering seen in the linked image (currently visible only in a not yet approved edit) is date imprinting, which should appear on the negatives as well.
The Olympus Trip AF shouldn't have this feature (according to @osullic), but misidentifying cameras is fairly common and there may have been an add-on for that camera that provided this.
If it's an add-on (replacement film door, for instance) there will be a separate battery that powers the clock and imprinter; removing that battery will stop the imprinter from putting dates (from twenty-some years ago, probably not settable past 2020) on your negatives. If you have a similar camera that has built-in date imprinting, it probably runs off the main camera battery that powers the auto-focus, zoom (if present), exposure, and flash electronics -- but someone who knows what they're doing could likely disconnect the wiring to the imprinter itself with similar effect.
These are not just random numbers – your camera has a feature where the date can be set, and this is imprinted onto the negatives, and consequently onto the photos. On the day that the above photo was taken, your camera was set to February 16, 1998.
The numbers are surely also on the negatives. Remember, because it's a negative, the "bright" numbers will appear dark on the negatives. I think you just need to look more closely, and you'll surely see them.
It's worth mentioning also that Olympus used the "Trip" name on many compact cameras. There was indeed one model called the "Trip AF" (introduced in 1984), but it didn't have the date imprinting feature, so I wonder if your camera is actually a different model.
If the numbers are on the negatives (and they are), then they will appear on the prints – that's just how it works. You may be able to disable the date imprinting feature on the camera itself. Best thing is to find and consult the manual for your camera. If it's not possible to switch the feature off (and I sympathise – I never liked this feature either), then unfortunately you might just have to learn to love it and live with it.
(Small piece of trivia: APS cameras had the ability to record the date on the magnetic portion of the film, rather than optically on the negative itself. Photo-finishers could then print the photo date on the rear of the print – a much better implementation.)