Seguimos la búsqueda en el bosque.

Here, seguimos means we continue / we keep up / we carry on with the search.

The verb seguir often means to follow, but with something like la búsqueda it can mean to continue or to pursue an activity.

So in this sentence, the idea is not usually we follow the search in a literal sense, but we continue the search.

Because seguimos has the same form in two tenses:

• present: we continue / we keep going
• preterite: we continued

So the form itself is ambiguous. Only the context tells you which one is meant.

For example:

• In a present-time situation: Seguimos la búsqueda = We continue the search
• In a past narrative: A pesar de la lluvia, seguimos la búsqueda = In spite of the rain, we continued the search

Spanish often leaves out subject pronouns because the verb ending already shows who the subject is.

• seguimos already tells you it is we
• so nosotros is usually unnecessary

You would add nosotros only for emphasis, contrast, or clarity:

• Nosotros seguimos la búsqueda, pero ellos se fueron.
  We continued the search, but they left.

Because búsqueda is a noun meaning search.

Spanish often uses a verb + noun structure where English does too:

• seguir la búsqueda = continue the search

A more verbal alternative would be:

• seguir buscando = keep searching

Both are possible, but they are structured differently:

• la búsqueda focuses on the search as an activity or operation
• buscando focuses on the action of searching

Spanish uses articles more often than English does.

Here, la búsqueda usually refers to:

• a specific search already known in context, or
• the search effort as a defined activity

So la is natural here. Spanish would generally not drop it in this sentence.

Seguimos con la búsqueda is very common and often sounds more natural in everyday speech for we continued the search.

So:

• Seguimos la búsqueda = understandable, but can sound a bit more formal, literary, or like we pursued the search
• Seguimos con la búsqueda = very common for we continued with the search

Also very natural is:

• Seguimos buscando en el bosque = We kept searching in the forest

They do not mean exactly the same thing.

• en el bosque = in the forest
  This gives the location of the search.
• por el bosque = through / around the forest
  This emphasizes movement through the area.

So en el bosque answers where?, while por el bosque suggests through what area?

Because Spanish only contracts these two combinations:

• a + el = al
• de + el = del

But:

• en + el stays en el

So en el bosque is correct, and there is no contraction there.

Búsqueda is pronounced roughly BOOS-ke-da.

The stress falls on the first syllable:

• BÚS-que-da

The written accent mark tells you that the stress goes there.

Without the accent mark, Spanish spelling rules would make you expect the stress elsewhere, so the accent is needed to show the correct pronunciation.