How can I research my family lineage back to Moldavia?

You learn how to research and prove first and that starts with you and the records you already have at home and in the homes of your living family, so your own birth cert proves you and your parents, their marriage cert will give you more information about them and their parents, all proved information and with just two real records you have a three generartion family tree and you continue in the same way proving and citing from real records... in the US once you get back to 1940 you can also view census returns and each 10 yrs back which shows you the family in the household and if someone migrated some census give the date of that and then you can look at passenger ship records to find who migrated and when...... family stories are nice to have however most are untrue, a few have 'some' truth but a story is not what you are researching it is the proved people and the real records they generated during their lifetime

Familysearch.com is the world's largest genealogy database, and it's free.

Go on Ancestry.com. I've been able to track my ancestors back to 1000 AD. Unfortunately, the records get a little hard to access in some countries, but if your ancestors were really prominent, they'll be there. Or try geni.com. If you have some names of grandparents and greatgrandparents, you may be able to track them there. 

Start learning how to do geneaological research. That means interviewing older family members, collecting lots of birth records, death records, immigration records, church records, military records, newspaper clippings, etc, etc, etc. It often takes years upon years to properly document all of that unless you have a relative who has done it. You'll only scratch the surface on the internet and you'll have to do significan't work overseas, or hire a professional researchewr.