Improvisations can be short lick/fill in a rock tune that lasts just a few beats or measures, or an extended jazz solo that lasts for multiple choruses. There are many different ways of choosing notes that will sound like they belong to the tune. What you choose will depend on the style, the speed of the harmony, and other considerations, but here are some options.

Generally for all these, there are two main factors to consider: the key of the tune and the chord of the moment.

1. Chord tones. Notes from the “chord of the moment” will always sound like they belong. So, if you’re in the key of C major and the chord G7 is active, you can certainly choose G, B, D, and F for your improvisation, and it will work There won’t be much melodic tensions, particularly if you choose a root or 5, but at worst, it will sound a little dull rather than “wrong,” so you might emphasize the 3, 7, and any specified chord tensions.  Spice it up with rhythm, including repeating notes and syncopation.
2. Non-specified chord tensions. On top of the chord tone, you can explore the 9, 11, and 13, as permitted diatonically by the key signature, and avoiding b9’s with other chord tones. So, on the chord G7 in the key of C major, you could try A, C, and E. The tension note C (11) forms a b9 with the chord tone B, but the other two will work fine, and add more interest than the chord root and 5 (which are likely to be sounded by other band members anyway). A great source of information about this is The Berklee Book of Jazz Harmony, by Joe Mulholland and Tom Hojnacki (edited by yours truly; here's my interview with the authors about it).

1. The song melody. Within a harmonic region, simply varying or rearranging melody notes is an easy source of soloing ideas. You can change their rhythm or order, repeat them, leave some out, put some extras back in, and so on, making it more or less recognizable as the melody per your taste.
2. Diatonic non-chord tones. A non-chord tone is a note that isn’t in the chord, but is diatonic to the key. There are some rules to follow here, as they will add tensions to the melody that needs to be resolved. Study counterpoint for that. But some good options are neighbor notes, which are diatonic notes adjacent to a chord tone, and passing tones, which are diatonic notes between two chord tones. So, on that G7 chord in C major, you could play the melodic figures D E D  (E is the lower neighbor) or F E D (E is the passing tone).
3. Chromatic non-chord tones. A chromatic non-chord tone is a half step away from a chord tone, such as a b3, b5, or b7 in a major key. These can be high tension, and so they have to be handled thoughtfully to sound good. A general rule is to avoid the sound of a conspicuous b9 interval with a prominent note. But especially if they are fleeting motions, handled as passing notes or neighbor notes (particularly as ornaments), they can be a nice way to add some color to a melody.
4. Pentatonic scales. Scales are a way of organizing larger sets of notes, and to start on this path, the first simple step beyond chord tones and their friends is the pentatonic scale built on the chord of the moment. There are various pentatonic scales; the most common are major (1 2 3 5 6) and minor (1 b3 4 5 b7). Choose the one that matches the chord of the moment, as informed by the key signature of the tune. For a G7 chord in C major, the major pentatonic on G (G A B D E) will sound right. For a Gmin7 chord in the key of C minor, the minor pentatonic G Bb C D F will work better.
5. Blues scales. The most common blues scale is like a pentatonic scale with an added b5 (in C minor, C Eb F Gb G Bb). The b5 is what makes it interesting—renegade, even, as it steps beyond the diatonic parameters of the key. In a sense, a blues scale gives you the best bang for your buck: the most interest out of the least complexity. There are other blues scales as well; including both the 3 and b3 is a common variant.
6. Chord scales. By chord scale, we mean the diatonic scale or mode that is made from the chord of the moment as informed by key signature. We saw this practice with pentatonic scales. So, for a G7 chord in the key of C major, you could augment the G B D F with notes from the key A C E, to arrive at a G Mixolydian mode. Any notes from G Mixolydian will sound convincing on that chord.
7. Common tones.  When changes are happening fast and you want to simplify, it can be helpful to consider all the chords in the region, and see what notes work in both places. So, if it’s changing between CMaj7(9) and Fmin6 every beat, look at the chord tones: C E G B D and F Ab C D, and spot what they have in common: C for sure.  Then, extend the idea to chord scales for each, and see what other notes are in common.
8. Instinct. At a point, you need to trust your subconscious to come up with ideas that will sound good, and not think deliberately about the theory of improvisation while you are blowing your horn. Your intuitive ear will develop, however, as you consciously learn different techniques, and thus help expand your concept of what lies within the boundaries of “good taste.”

Each of these topics is worth significant study in its own right. The math can get as easy or as complex as you like. Often, a complicated flurry of notes winds up being actually highly forgettable, and so, if you’re going for art, rather than theory exercises, it is often most effective to stick with gestures based on the melody, to remember rhythm, and to be unafraid to repeat notes, or hold notes, and otherwise try to create something that can stand on its own, rather than simply “make the changes.”

A common misconception of music education is that students think teachers are presenting “the right” way to do something. Rather, though, education is really about exposure to different possibilities, with the expectation that in the long term, the study will result in an artist with a rich creative background from which to forge new directions. Often, higher education in music theory is in fact a methodical way to teach musicians how to get their freakiest freak on.

Dip your toe in that, but then step back and regroup, and find your own voice and consider what makes sense to you. Ultimately, that is about self trust.