The framework for music is math, especially the harmonic series. For rhythm, harmony and melody it is the mathematical relationship that creates the patterns that our brains look for when searching for meaning. Harmonic Polyrhythms Explained! by Adam Neely does a great job of explaining and illustrating this in a way you can hear.
The meaning that our brains associate with these patterns is not just about math, however; rather we build up associations between certain math patterns and emotional or cognitive meaning. How much the association between specific patterns and meanings is objective or subjective is debated, but most will agree that there is an association and that it is fundamental to our experience of music.
Choice of Tonal Center
From a pure math perspective, in a just intonation system the choice of tonic does not matter. All the ratios and patterns can exist equivalently based on any of the 12 notes. It is all about the relationships between the notes, so if we shift the tonal center, as long as we shift all the other notes by the same amount the effect on the listener should be the same.
In practice this is not quite true for two main reasons: the mechanics of the instruments we use and the perception of the listener. Many instruments in use today are not perfectly just intoned (e.g. all fretted string instruments like guitar). This will cause some keys to have different ratios than other keys, which means we cannot shift the key and get an identical effect. The listener's history of music experience will have a significant effect on how a key is interpreted. Often preconceptions can be overcome quickly but any patterns in the music the listener has heard to date will color the perception of the tonal center used. Other concerns related to how high or low (in frequency) or the timbre of the instrument in differing keys can affect how the music is perceived both in the mechanics and the listener's expectations.
The mechanics of playing your instrument is different in each of the 12 tonal centers. This is a problem that you can address individually by learning the note patterns of (ideally) all the keys. In this type of practice it is not necessary to think too much about the modes or music theory so much as developing the muscle memory for each pattern. Not all 12 patterns are equally common, so think about what other instruments you want to play with and the most common keys in the music you want to play to guide some priority in developing your mastery of each pattern on your instrument.
The seven modes in one key are representative of those seven modes in any key, so learning study of these modes in one key will be applicable to all twelve tonal centers.
The modes can be thought of as modifications of major. Lydian is one less minor than major, i.e. you add one sharp, which is the four of the scale. In the more minor direction (with scale note): mixolydian (7), dorian (3), minor (6), phrygian (2), locrian (5). You could think of them like this: +4, 0, -7, -3, -6, -2, -5. I find that organizing them in my mind this way is helpful. If you were listening to music you could start by finding the tonic and then look at each of these intervals to determine what mode the music is in.
Adam Neely does a better job than me bringing all these ideas together in his video Why is major "happy?"