Is it normal to have ups and downs during the first weeks of antidepressant treatment?

Yes. Antidepressant medication takes 4-6 weeks to become effective.
Depends. If your ups and downs become that you were having before starting an antidepressant become intensified or more frequent after starting an antidepressant, that may suggest that you have underlying bipolar disorder and need to reevaluate your medication regimen.