Every question I have seen includes flurry of blows, or readied actions, or whatever. But I am talking about level 1 monk, using just an attack action, and just the martial arts feature. You can attack, and you can make an unarmed attack as a bonus action, using the Martial Arts feature.
Can the monk move between these attacks?
When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff, you can also make an unarmed strike as a bonus action, assuming you haven’t already taken a bonus action this turn. - PHB pg.78
You choose when to take a bonus action during your turn, unless the bonus action's timing is specified, and anything that deprives you of your ability to take actions also prevents you from taking a bonus action. - PHB pg.189
You can move before or after an action:
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. - PHB pg.189
You can break up your movement:
You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet. - PHB pg.190
And you can move between attacks:
If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks. For example, a fighter who can make two attacks with the Extra Attack feature and who has a speed of 25 feet could move 10 feet, make an attack, move 15 feet, and then attack again. - PHB pg.190
This would all seem to imply that you can indeed move in between these. Bear in mind, though that the last point is talking about attacks within an attack action, whereas these are a separate action and bonus action, so we must consider: if "you choose when to take a bonus action during your turn, unless the bonus action's timing is specified" and "when you use the Attack action with an unarmed strike or a monk weapon on your turn" is considered specifying timing rather than just conditions, then you must take the bonus action straight away.
Based on the ruling that the Shield Master feat can only be used after the attack action and keeping in mind that the wording there is simply
If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield." - PHB pg.170
and the word in the Martial Arts feature is when (much more temporal in it's common use) it would seem reasonable to rule that these two attacks must be taken together. That said, especially before flurry of blows becomes available at the next level, allowing movement between them does not seem at all unbalanced to me.
On whether the unarmed attack is a weapon attack, this is most clearly indicated here:
Whenever you take an action that includes more than one weapon attack, you can move between those attacks if you have enough movement to do so. This rule applies to bonus actions, which are a type of action, and to unarmed strikes, which are weapon attacks. - Jeremy Crawford, Lead Rules/PHB Designer
As ravery noted, however, the fact that an unarmed strike is considered a melee weapon attack does not mean that the monk's fist is considered a melee weapon in the rules. This may be relevant for abilities/effect that specify "when you attack with a melee weapon" rather than "when you make a melee weapon attack," though it wouldn't affect whether you can move between such attacks or their related actions.
Can the monk move between these attacks?
Jeremy Crawford, rules designer for D&D 5e, confirms that this is possible here:
Whenever you take an action that includes more than one weapon attack, you can move between those attacks if you have enough movement to do so. This rule applies to bonus actions, which are a type of action, and to unarmed strikes, which are weapon attacks.
Though the Twitter user he's responding to asked about Flurry of Blows, his answer is more general and applies to any and all situations involving multiple attacks (unless the feature specifically says otherwise).
Regarding the ability to move between attacks in general, see the section of the basic rules titled "Breaking Up Your Move":
You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet.
Moving Between Attacks
If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks. For example, a fighter who can make two attacks with the Extra Attack feature and who has a speed of 25 feet could move 10 feet, make an attack, move 15 feet, and then attack again.
The answer to movement, is no. The monk will attack with the monk weapon (if completely unarmed, consider this the dominant hand), and follow suit with an(other) unarmed weapon. The second unarmed does count as a weapon attack (weapon in this case is a fist).
