This is going to be a long winded answer to a seemingly simple question. I think to some degree your question isn't fair because you're asking us to pick one of 2 things that actually complement each other rather than compete.
In short you need to do both, memorize the patterns and translate them, and memorize how the different patterns are connected.
There are at least two levels of learning modes. One is the development of the haptic connection to the patterns on the instrument in some tuning (let's assume standard). The other is learning that, in theory, the diatonic modes are all the same. "Best way to learn..." may not be a fair question. People can only share what works for them and that may not work for you.
I would say that as a guitarist the haptic connection has to take precedence (probably true of any instrument). If you know all the theory in the world your hand won't respond to what your ear tells you to do unless the patterns are muscle memory. The guitar is a fascinating instrument in that it has transnational invariance. If you know your patterns in Amaj (relative to the 5th fret for example) you can move those patterns up to Bb or C and they are all the same. This can make the job a little easier. Many shredders (in videos from the 80s etc) advocate practicing patterns chromatically up the neck. I personally think we can do better by jumping thirds or going through the circle of 4ths as we progress. The guitar will start to feel different in positions that are far apart. I personally like to drill patterns (scale, arpeggio, changes, etc) in F or G at the 1st or 3rd fret (but not open position), then at the 5th or 7th fret, again up near 12th, the 15th or high if necessary. This gives you enough coverage to feel your way through the other keys. Open string patterns need to be practiced independently as they tend to use different fingerings. It seems to me that modern guitarists have abandoned open string mode patterns but they are important. I personally like to go through a cycle rather than chromatically. This advice applies to all modes (diatonic, related to melodic minor, pentatonic, etc). Drill until the finger cannot be confused.
Connecting modes to chords. The end goal has to be using the modes to solo over changes or understanding these patterns for composition etc. To that end it is very helpful to practice those modes that fit easily over chord pattern on the guitar (this is a guitar-centric practice method as it appeals to the different hand shapes). There are 5 basic ways to play the major and minor (and seventh) chords, that are all similar to the open string chord forms (E, A, D, G, and C). The forms are usually referred to as the C-A-G-E-D system. In a nut shell Phygian or Lydian mode fits over (or matches) that C-form of any chord, Locrian or Ionian fits the E-form, Dorian the D-form, Aeolian the G-form, and Mixolydian the A-form. This is not a statement of music theory, like what mode matches what triad or 7th chord in a set of changes. This is a statement about the physical form of the mode and it's connection to the root of a key. By practicing 5 of the 7 modes that match the chord forms you will eventually develop an immediate physical connection between a "feeling" in your hand shape and the modes of the key closest to your fingers. As musicians the connection should happen in the ear first (and it does with practice). But this physical connection is invaluable for getting your hand to connect to your ear w/o having to overthink. Along these lines I like to practice modes as follows. Start with G Ionian, followed by G mixolydian, G Dorian, G Aeolian, G Phygian. This pattern takes you through the following key changes: G, C, F, Bb, Eb. In other words the circle of fourths. Keep going all over the neck.
Connecting modes to each other. Once you've got the patterns down and you know how they are related to an open string chord form connect them by connecting the 5 forms of the I chord in a single key. For example starting in G, G E-form at the 3rd fret followed by A dorian at the 5th fret (G D form), then B Phrygian at 7th (G C-form), Mixo at the 10th fret (G A-form), and finally Aeolian at the 12th fret (G G-form). You can continue on to the octave at the 15th fret. Notice that some modes are missing. It's not that they are not important but in terms of finger patterns and haptic memory they are not very different for the guitarist. This applies to the melodic minor modes as well.
Find all chords in a mode. Connecting a mode to a chord (like Dorian to the ii-7) can be inverted and one can find ALL chords in a key in each and every mode of that key. Another post.