Prefiero ir a pie a la biblioteca cuando hace buen tiempo.

After preferir, Spanish normally uses a second verb in the infinitive directly, with no extra preposition:

• Prefiero ir = I prefer to go
• Prefiero comer = I prefer to eat
• Prefiero quedarme = I prefer to stay

So preferir + infinitive is the usual pattern.

A pie is a fixed expression meaning on foot or walking.

Examples:

• Voy a pie = I go on foot / I walk
• Está a diez minutos a pie = It’s ten minutes away on foot

Even though pie literally means foot, the whole phrase a pie should be learned as one expression.

They are both spelled a, but they do different jobs here:

• a pie: part of a fixed expression meaning on foot
• a la biblioteca: the normal preposition to, showing destination

So in:

• ir a pie = to go on foot
• ir a la biblioteca = to go to the library

You can think of the sentence as:

• I prefer [to go] [on foot] [to the library] [when the weather is good].

Spanish usually uses a before a destination with verbs like ir:

• Voy al supermercado
• Vamos a casa
• Iban a la estación

Here, biblioteca is a feminine singular noun, so a + la biblioteca = to the library.

Note:

• al = a + el
• but a la stays separate

So:

• a el museo becomes al museo
• a la biblioteca stays a la biblioteca

In Spanish, weather is often expressed with hacer in fixed phrases:

• Hace frío = It’s cold
• Hace calor = It’s hot
• Hace sol = It’s sunny
• Hace buen tiempo = The weather is nice / It’s good weather

So hace buen tiempo is the natural idiomatic way to say this.
Using es or está here would not sound natural.

Buen is the shortened form of bueno before a singular masculine noun.

• un buen día
• buen tiempo
• un buen libro

Since tiempo is masculine singular, bueno becomes buen before it.

Also, buen tiempo is a common expression meaning good weather / nice weather.
If you said tiempo bueno, it would sound unnatural in this context.

This is called apocope: some adjectives lose their final vowel before a singular masculine noun.

So:

• bueno → buen before a masculine singular noun
• malo → mal before a masculine singular noun

Examples:

• un buen amigo
• un mal día
• buen tiempo

But if the adjective stands alone, you use the full form:

• El tiempo es bueno

Yes, Spanish word order is flexible, and both are possible:

• Prefiero ir a pie a la biblioteca
• Prefiero ir a la biblioteca a pie

Both are understandable and natural.

The version with a pie before a la biblioteca slightly emphasizes how you go before where you go.
The other version may feel a bit more straightforward: destination first, then manner.

In everyday Spanish, both can be used.

Prefiero already means I prefer, because the verb ending shows the subject.

• prefiero = I prefer

Spanish often drops subject pronouns when they are not needed:

• Prefiero ir a pie is completely normal.
• Yo prefiero ir a pie is also possible, but yo adds emphasis or contrast.

For example:

• Yo prefiero ir a pie, pero él prefiere conducir.

Prefiero is the present tense, first person singular, of preferir.

Conjugation:

• yo prefiero
• tú prefieres
• él/ella prefiere
• nosotros preferimos
• vosotros preferís
• ellos/ellas prefieren

Notice that preferir is a stem-changing verb:

• e → ie in most forms
• but not in nosotros and vosotros

So:

• prefiero, prefieres, prefiere
• preferimos, preferís
• prefieren

In weather expressions like hace frío or hace buen tiempo, Spanish uses hacer in the third person singular:

• Hace frío
• Hace calor
• Hace mal tiempo
• Hace buen tiempo

This is an impersonal use, similar to English it is... in weather expressions.
So hace is the normal form, even though no specific person is doing the action.

Yes, you could say:

• Prefiero caminar a la biblioteca cuando hace buen tiempo.

This means I prefer to walk to the library when the weather is nice.

The difference is:

• ir a pie = to go on foot
• caminar = to walk

They are often very close in meaning here.
Ir a pie is a very common expression, and caminar is also perfectly natural.

Biblioteca means library, a place where books are kept for reading or borrowing.

A bookshop / bookstore is:

• librería

This is a very common false friend for English speakers:

• biblioteca = library
• librería = bookshop/bookstore

So in this sentence, la biblioteca definitely means the library.