This is a good question, and I think if you keep going down and leave our solar system, go through the Kuiper Belt, then the Oort cloud, then leave our Galaxy then say goodbye to our local group you’d just keep finding empty space for a long long time.
I don’t think anyone knows for sure whether if you kept going forever you’d ever encounter some form of ‘edge’.
I do know it’s complicated by the fact that space itself is expanding (kinda like if you blow up a balloon, it’s surface ‘expands’).
Philip is right, the expansion of the universe gives this an extra twist, especially because the expansion is accelerating. If the acceleration continues, you will not even able to reach all the objects you can see on the sky. This is because the most distant objects are moving away from us faster than the speed of light, which is the fastest speed at which we can travel. Therefore we cannot reach them. I am afraid this means that you will not find anything you cannot see from the Earth…