Learn Turkish: Plajda yüzüyorum ve dinleniyorum. - I'm swimming and resting at the beach.

Plajda uses the locative suffix -da, which means "at", "on", or "in" a place. If you said plaja, it would mean "to the beach". Since you are already at the beach swimming and resting, you need the "at" ending (-da), not the "to" ending (-a).

Why is there an '-ü-' in 'yüzüyorum' but an '-i-' in 'dinleniyorum' before the '-yor' ending?

This is because of vowel harmony. The present continuous suffix -yor requires a connecting vowel that matches the verb root's last vowel. The verb root yüz (swim) has the vowel ü, so it takes -ü-. The verb root dinlen (rest) has e as its last vowel, which pairs with -i-.

Why is the ending on 'plaj' exactly '-da' and not '-ta' or '-de'?

First, it uses an a because the only vowel in plaj is a (following the a/ı/o/u vowel harmony group). Second, it uses d instead of t because j is a voiced consonant. The harder -ta ending is only used after voiceless consonants (like p, ç, t, k, s, ş, h, f).

Why do both verbs end in '-um'? Can I just attach it to the final verb?