Learn Hardcore German: Wir gehen den Fluss entlang und erkennen die alte Brücke sofort. - We walk along the river and recognize the old bridge immediately.

Because entlang is (most commonly) a postposition that follows its noun and governs the accusative. So you say den Fluss entlang. If you place entlang before the noun, formal written German prefers the genitive: entlang des Flusses. You will also hear a colloquial/regional dative in speech (entlang dem Fluss), but it’s better to avoid that in standard German.

Two safe, standard options:

den Fluss entlang (accusative, postposition; very common)
entlang des Flusses (genitive, preposition; more formal)

There’s also a very common alternative with another preposition:

am Fluss entlang = an dem Fluss entlang (dative with an, then entlang placed at the end)

Is there any difference between den Fluss entlang and am Fluss entlang?

Semantically, both mean “along the river,” and in everyday use they are interchangeable. Subtle tendencies:

den Fluss entlang often focuses on moving along the length/line of the river itself.
am Fluss entlang emphasizes being alongside the river (by the riverbank) as you go.

In practice, native speakers freely use both.

Is entlang a separable verb prefix like mit-? How does word order work?

In your sentence, entlang is a postposition, not a verb prefix: Wir gehen den Fluss entlang.

However, there is also the particle verb entlanggehen (“to go along”), typically used with an + Dativ:

Main clause: Wir gehen am Fluss entlang.
Subordinate clause: …, dass wir am Fluss entlanggehen.

With the postposition variant, entlang stays next to its noun phrase even in subordinate clauses:

…, dass wir den Fluss entlang gehen. Do not say: ✗ …, dass wir entlang den Fluss gehen.

Why is sofort at the end? Where else can it go, and does the meaning change?

German adverbs are flexible. All of these are correct:

Wir erkennen die alte Brücke sofort. (neutral)
Wir erkennen sofort die alte Brücke. (slight emphasis on the immediacy)
Sofort erkennen wir die alte Brücke. (strong focus on “immediately”)

The meaning stays the same; word order mainly affects emphasis/information structure.

Why use erkennen instead of sehen here?

sehen = to see (mere visual perception)
erkennen = to recognize/identify (you realize what it is)

So wir erkennen die alte Brücke sofort means you identify it at once, not just that you saw it. Related verbs:

bemerken = to notice
entdecken = to discover (something previously unknown)
Pattern: etw. an + Dativ erkennen (e.g., Wir erkennen die Brücke an ihrer Form.)

Why is there no comma before und?

Here, und links two verb phrases with the same subject (wir). In such cases, German does not use a comma:

Wir gehen … und erkennen …

A comma can be used between two independent main clauses (e.g., with different subjects), but in this sentence it would be wrong to insert one:

✗ Wir gehen den Fluss entlang, und erkennen die alte Brücke sofort.

Can the subject wir be omitted in the second part after und?

Yes. When two predicates share the same subject, German normally states the subject once and omits it in the second part:

Wir gehen den Fluss entlang und [wir] erkennen die alte Brücke sofort.

If the subject changes, you must state it:

Wir gehen den Fluss entlang, und die Kinder erkennen die alte Brücke sofort.

Why is it die alte Brücke (and not den alten/die alten Brücke)? What’s going on with the adjective ending?

Brücke is feminine: die Brücke.
As a direct object, it’s in the accusative, but feminine nominative and accusative are both die.
With a definite article (die), the adjective takes the weak ending -e: die alte Brücke.

Compare:

Masculine accusative: den alten Fluss (if you added an adjective)
Feminine accusative: die alte Brücke
Plural (any case with definite article): die alten Brücken

What are the genders of Fluss and Brücke, and how do the articles change?

der Fluss (masculine) → accusative singular: den Fluss
die Brücke (feminine) → accusative singular: die Brücke

You largely have to memorize grammatical gender in German.

Does entlang behave like the two-way prepositions (Wechselpräpositionen) with movement vs. location?

No. entlang is not one of the standard two-way prepositions like an, auf, in. As a postposition, it takes the accusative by default (den Fluss entlang). The movement/location distinction applies to prepositions such as an:

Location: am Fluss = an dem Fluss (dative)
Direction: an den Fluss (accusative)

How could I express “upstream” or “downstream” here?

Use:

flussaufwärts = upstream
flussabwärts = downstream

Example: Wir gehen flussabwärts und erkennen die alte Brücke sofort.

Are there other natural ways to say “immediately” besides sofort?

Yes:

gleich (often “soon” in some regions; can mean “immediately” in others—potentially ambiguous)
auf der Stelle (on the spot)
umgehend (promptly; more formal)
auf Anhieb (right away/at the first try), e.g., Wir erkennen die Brücke auf Anhieb.

How would I say this in the past?

Two natural options:

Conversational past (Perfekt): Wir sind den Fluss entlanggegangen und haben die alte Brücke sofort erkannt.
Simple past (Präteritum): Wir gingen den Fluss entlang und erkannten die alte Brücke sofort. In everyday speech, Perfekt is more common; Präteritum is frequent in writing and with some common verbs.

Can I use laufen instead of gehen?

Be careful:

In standard German, laufen usually means “to run.”
In some southern regions, people may use laufen to mean “to walk,” but this can confuse non-locals.

Neutral and safe for “to walk” is gehen:

Wir gehen den Fluss entlang … is the best default.