Learn German: Der See war sehr kalt, aber der Fluss war kälter. - The lake was very cold, but the river was colder.

In German, when you make the comparative form by adding -er to an adjective, many short, one-syllable adjectives with an a, o, or u also get an umlaut. That is why kalt becomes kälter.

Is der See always the lake? I remember learning that it means the sea.

It depends on the gender! Der See with the masculine article means the lake, which is used here. But die See with the feminine article means the sea or ocean. It is a classic trap for learners, so always memorize the gender with the noun.

Why is the word order normal after aber? Shouldn't the verb war move?

Aber is a coordinating conjunction, which means it does not push the verb to the end of the sentence. It counts as position zero, so der Fluss is in position one, and the verb war stays safely in its normal second position.

Could we use the spoken past tense (Perfekt) here instead, like ist kalt gewesen?

You could theoretically say ist kalt gewesen, but for the basic verbs sein and haben, German speakers almost always use the simple past forms war and hatte even in casual spoken conversation. It is much shorter and more natural.