The rules unfortunately use natural language and seemingly contradict themselves. Ask the GM
Reading with natural language, the rules seem to state that both bonus actions and reactions are actions
Some rules quotes on bonus actions and reactions:
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.
Certain special abilities, spells, and situations allow you to take a special action called a reaction.
I would absolutely assume this means bonus actions and reactions are both actions. If an item has a special or additional fee, in either case I would agree it has a fee. If something is a special or additional action, I would naturally conclude it must also be an action.
Yet when something requires your action, it doesn't mean your bonus action or reaction
Countless features throughout the game use phrases such as "as an action" or "using your action" or similar and these refer to your actual, big action. Similarly there is the fighter's Action Surge:
On your turn, you can take one additional action.
This only lets you take an additional action and does not allow for a second bonus action (or reaction) despite bonus actions literally being called "an additional action". The SAC settles this clearly:
Q. Does the fighter’s Action Surge feature let you take an
extra bonus action, in addition to an extra action?
A. Action Surge gives you an extra action, not an extra bonus action.
Nonetheless there are times the rules refer to an action and do include bonus actions
Further complicating this are One With Shadows and an SAC entry concerning it:
When you are in an area of dim light or darkness, you can use your action to become invisible until you move or take an action or a reaction.
Q. Does using a bonus action break invisibility from a warlock’s One with Shadows invocation?
A. Taking a bonus action breaks the invisibility of a warlock’s One with Shadows. A bonus action is an action.
So here we see "take an action" include bonus actions because "a bonus action is an action". And yet, Action Surge, when letting you take "an additional action" apparently does not allow for an additional bonus action. Furthermore, why is there this rule, if bonus actions really are actions:
anything that deprives you of your ability to take actions also prevents you from taking a bonus action
Perhaps the rule is just redundantly repeating information. Or perhaps it's clarifying that, unlike with Action Surge, in these specific scenarios, when something affects your action it also affects your bonus action.
Meanwhile, reactions are not meant to be considered actions
We should also note that while bonus actions are not called out explicitly in One With Shadows, reactions are. The practice of calling out reactions but not bonus actions is extremely common throughout the rules, such as in the Incapacitated condition, and this seems to indicate that reactions are not actions. According to Crawford, and the most upvoted answer to this related question this is the case:
Actions and reactions are different. If an effect, like the haste spell, shuts off one, it doesn't necessarily shut off the other.
Actions and reactions are different.
This is despite the fact that reactions are described nearly identically to bonus actions, which somehow are actions (well... except when they aren't, like Action Surge).
You also don't provoke an opportunity attack when you teleport, or when someone or something moves you without using your movement, action, or reaction.
So what happens if you move using your bonus action? The written rules are really a lot of a mess. At times they assume bonus actions are actions (One With Shadows), at times they assume they are not ("as an action" and Action Surge), and at times they assume reactions are not actions (listing both out in various features). Meanwhile, the text that supports that bonus actions are actually actions is almost identical to the description of reactions, so why would we interpret these phrases to have completely different meanings? (This sheds more confusion on why bonus actions are actions and thus whether they are actions at all).
I don't see any real way to reconcile all of these rules in a coherent manner. Bonus actions simultaneously do and do not count as actions, and I don't see any easy way to know when each is the case. The writers unfortunately used natural language, which didn't help here, and picking through them for the RAW gets mixed results. I say, ask the GM.
At my own tables, using your bonus action to move provokes opportunity attacks, whether this is a houserule or not, I could not say.