Dear Joanna, There are good Irish genealogy sites on the internet.There is one on Rootsweb and 2 here on Genforum.2 others are Fianna and Genuki.
If you know about when your Grandmother arrived in the US, you can find her immigration record (arrival document) for free on Ellis Island or on Castle Garden.Castle Garden processed people arriving in New York before 1892.Ellis Island took over in 1892 and has records from other ports than just New York.Ellis Island and Castle Garden are free sites, unless you want pictures of the original arrival documents or of the ship they traveled on to get here.Ancestry is not free, except for 2 week free trial, but it lets you download or print out the original arrival documents. You can access it for free at most public libraries.
I search for immigrant ships on the internet, rather than paying for a photo, because some sites have a huge collection of ships' photos, and you can download them.One of the biggest is The Ships List.One with most of the immigrant ships is Norway Heritage.(The Norwegians owned a lot of the immigrant ship lines.)
A lot of Irish arrived in Boston, rather than New York, or came in through Canada, because the British controlled both Ireland and Canada for much of the immigration years.Their immigration records are harder to find than the New York arrivals.It's a good idea to look at Canadian border crossing records, when you can't find someone.
You should try to get the immigration records first, because they tell the arrival date, the date they departed the home country on, and the name of the ship.They often tell where the person was from by city and county or province. They can also have the name of a contact person, usually a parent, in the home country.Then you can use Fianna or Genuki more effectively.
Ancestry has some genealogy records from England and Ireland and Canada if you get the international subscription, which is more expensive.I don't know if libraries have the international records, but I would check them first, because the library's access is free.
I hope this information will help you find your Grandmother's history. Sincerely, Kate Hagel