The Einstein rings and proper motions of nearby stars tend to be large. Thus, every year some foreground stars within a few hundred parsecs of Earth induce gravitational lensing events in background stars. In some of these cases, the events may exhibit evidence of planets orbiting the nearby star. In fact, planets can even be discovered during relatively distant passages. Here, we study the lensing signatures associated with planets orbiting nearby high-proper-motion stars. We find the following. (1) Wide-orbit planets can be detected for all distances of closest approach between the foreground and background stars, potentially producing independent events long before and/or after the closest approach. (2) Close-orbit planets can be detected for intermediate distances of closest approach, producing quasiperiodic signatures that may occur days or weeks before and after the stellar-lens event. (3) Planets in the so-called 'zone for resonant lensing' can significantly increase the magnification when the distance of closest approach is small, making the stellar-lens event easier to detect, while simultaneously providing evidence for planets. Because approaches close enough to allow planets to be detected can be predicted, we can plan observing strategies to take advantage of the theoretical framework built in this paper, which describes the sequence of expected effects in terms of a sequence of detection regimes.