If you have a shared bank account, start paying attention at what items appear on the statement. Is there anything suspicious? Purchases from stores you have never heard of? Payments for a phone line that you know nothing about or for hotels you never stayed at? These are all useful breadcrumbs to start a serious investigation.
If you manage to sneak into your partner's email account, try to locate any unfamiliar bank statement, phone line monthly summary, credit card bill and so on. Many experienced cheaters are well versed at covering their tracks with side bank account and a second, secret phone.