It depends on how much time you're going to have. With only 10 days, it's better to reach out and probably decide on a clinic and book surgery.
The first thing is to find out what the treatment cycle is like for what you want to have. Example:
Rhinoplasty treatment cycle is about 8 days.
Prep and consult day: 1
Day 2 – Surgery
Day 3 – Follow-up
Day 4 – Recovery / rest
Day 5 – Follow-up
Day 6 – Recovery / rest
Day 7 – Recovery / rest
Day 8 – Stitch removal and post consult
Now factor in travel days (arrival / departure), that's about 2 days.
That's a 10-day trip if you already have the surgery day locked in.
If you plan to consult with a few places first and then decide, you have to factor in consult days. Maybe you need 2 to 3 days to consult. Then factor in the delta or difference between your decision and the next available surgery date. Let's assume this can range between 3 to 7 days.
Now factor in travel days (arrival / departure), that's about 2 days.
So if you want to consult first and then decide, on the low end you need about 14 days, and on the high end about 19 days.