Check this list: https://www.adr.com/Markets/CountryProfile?code=ISR .
TASE is denominated in NIS, and most, if not all, companies that are traded there are tied to Israel and its currency. You can try going with the global companies (TEVA, NICE etc) which although are traded on TASE and are Israeli, have large global presence and are not dependent on Israeli market or currency (of course they are, but not tightly).
In Israel there's tons of various mutual funds that you can invest through without hassle, and save yourself the dealing with foreign exchanges by paying the slightly higher management fees and getting slightly lower income and lack of flexibility and control.
Or you can convert a large sum of NIS to USD (through an investment house it will be much cheaper than through a bank, I'm talking about commissions of $25-45 including the SWIFT fees here), and open an investment account in a US firm (or Israeli investment house that allows trading abroad). That is actually pretty simple and straight forward.
Oh, and bottom line - don't do your investments in a bank. Not in Israel, and not everywhere else. Banks are not for investments, banks are for dealing with cash and short term savings.