I have the sequence from an exon. Does the reading frame always start at the first nucleotide of the exon for when I want to translate it to an amino acid?
No, it doesn't. There's no way to know just by looking at an exon what frame is the right frame for translation.
If you know the length of your 5'UTR you can calculate the frame from the exon sequence, right?
If you know what position that exon sits in the whole transcript, then sure. But you can't know that from the sequence of an exon alone.