Learn Russian: Зимой мы берём лыжи.

Зима is the dictionary form for 'winter', but to say 'in winter' or 'during winter', Russian uses the instrumental case. For feminine nouns ending in -а, the instrumental ending is -ой, making it зимой. This pattern is used for other seasons too, like весной (in spring).

The infinitive is 'брать' (to take). Why does it change to 'берём' here?

The verb брать has an irregular stem change in the present tense. Instead of keeping the -а-, the root changes to бер-. It then takes standard Type 1 endings: я беру, ты берёшь, он/она берёт, мы берём, вы берёте, они берут.

Why is 'лыжи' spelled with an 'и' at the end instead of 'ы'?

Лыжи is the plural of лыжа (ski). Normally, feminine nouns ending in -а change to -ы in the plural. However, the Russian spelling rule states you can never write the letter ы after ж (as well as ш, ч, щ, г, к, or х). Because of this rule, you must write it with an и.

The English translation says 'we take our skis', but there is no word for 'our' in the Russian sentence. Why?