Learn Turkish: Sizi gördüm, çok memnun oldum.

The -i is the direct object ending (accusative case). Because 'you' are the specific target of the action, the verb görmek (to see) requires this ending on the pronoun siz.

Does sizi mean this sentence is spoken to a group of people?

It could be, since siz is the plural 'you'. However, siz is also used for a single person to show respect or formality. This sentence is a polite greeting, so it is very likely being said to just one person.

Why do the past tense endings look different between gördüm and oldum?

This is due to vowel harmony. The past tense suffix changes to match the vowel in the verb root. Since gör has an 'ö', it takes -dü. Since ol has an 'o', it takes -du. Both end with -m to show that 'I' did the action.

The English translation says 'I am pleased', but isn't oldum in the past tense?

Yes! Oldum literally means 'I became' (from the verb olmak). In Turkish, to say you are pleased, you express that you just entered that state. So memnun oldum literally means 'I became pleased', which translates naturally to 'I am pleased'.