It should be what you see in bold below. The underlined terms are the corrections.
El domingo fui a una boda y mis amigos estuvieron allí. El martes fuimos a patinar juntos después de la escuela. Mi mejor amigo comenzó las clases de guitarra ayer. Convencí a mis amigos unir a mi estudio de baile, después de la escuela vinieron a bailar conmigo.
The non-conjugated verbs used were
- ir = to go
- estar = to be
- comenzar = to start, to begin
- convencer = to convince
- unir = to join
- venir = to arrive
Their conjugations in the preterite tense are
- fui = I went
- estuvieron = they were
- fuimos = we went
- comenzó = he started, he began
- Convencí = I convinced
- no conjugations needed for "unir", since you're saying "to join"
- vinieron = they arrived
In short, here are the changes you need to make
- You need to change estaban to estuvieron
- Replace "para que se unieran" with "unir"
- Change vienen to vinieron
If you are curious what this translates to in English, then it would be
"On Sunday, I went to a wedding and my friends were there. On Tuesday, we went skating together afterschool. My best friend started guitar lessons yesterday. I convinced my friends to join my dance studio, they arrived after school to dance with me"
Side note: estaban is in the imperfect past tense, which is slightly different from the preterite.