Each of the verbs that you've mentioned has several (or many) meanings, but let's focus on the contrasts.
Caminar means to walk, i.e. to move with one's feet. There are figurative uses of this verb that don't imply that, but for the most part caminar means your legs are moving. It's (generally) an intransitive verb, so you can say
- Ella camina ("She walks"), no specific beginning or end implied, no direction given;
- Ella camina hacia el puerto ("She walks toward the port"), which indicates a direction and/or an intended goal;
- Ella camina hasta el puerto ("She walks [up] to the port"), specifying the endpoint of the movement;
- Ella camina desde su casa hasta la escuela, with both starting and ending points; etc.
There are some transitive uses of caminar: you can say caminar las calles, which has a connotation of work or effort or dedication (if not you'd say caminar por las calles). Note the direct object of caminar refers to the place being walked upon (e.g. the streets). There's no direct equivalent to the English "to walk the dog"; you say (sacar a) pasear al perro.
Andar can often be substituted freely for caminar (see this answer), but it's more generic, as it actually serves for any manner of motion (if you don't specify, it's understood that the motion is on foot). It's also intransitive.
There are some usages where andar means about the same as estar; these have nothing to do with motion but more with the idea of imprecise location, "to be around".
Andar also means "to work properly", "to function", generally said of machines or mechanisms.
Pasear means "to go around for fun or distraction". It doesn't imply any specific means of locomotion. If not specified the default is on foot. You can pasear a pie or pasear en automóvil. This verb is intransitive if it's the subject that does the movement (Ella pasea por el parque), transitive if the subject makes the object move (Ella pasea a los niños). Related to this difference in transitivity, compare the idioms:
- Sacar a pasear or sacar de paseo = "to take someone out for a walk" (it sounds a bit as if by force);
- Salir a pasear or salir de paseo = "to go out for a walk"
Note paseo can mean both "a walk, a stroll" and "an excursion". The idea in any case is that pasear or a paseo don't usually have a specific goal.
Pasar has several related meanings: "to go past", "to go through", "to go somewhere in the midst of doing other things", etc. In this sense it's intransitive. By itself it means "to pass, to go by":
- Ella pasó. = "She went by."
- Ella ya pasó. = "She went by already; she was already here and is elsewhere now."
In general you use pasar followed by a prepositional phrase.
- Ella pasó por mi casa. = "She called in at my house; she came by and then left."
- Ella pasó frente a mi casa. = "She went by the front of my house (she didn't call in but just went past)".
- Ella pasó bajo el puente. = "She crossed behind the bridge."
There are transitive usages of pasar. The ones that refer to movement mean "to make something go along/across/through something else"; mark the difference:
- Ella pasó su mano por la pared. = "She slid her hand across the wall."
- Ella pasó su mano por la tubería. = "She slid her hand in through the pipe."
Recorrer is always a transitive verb. It means "to go around", with the connotation of visiting and observing or inspecting.
- Ella recorrió las instalaciones. = "She went around (for a tour of) the facilities."
- Ella recorrió todo el Lejano Oriente. = "She went around (travelled) the whole Far East."
Note there's no preposition; you can't say *Ella recorrió por el país, for example.
For reference on the matter of motion verbs, this Wikipedia article on verb framing might be useful to you. It's not a reference of Spanish verbs but it highlights an important difference with motion verbs in English.