It will be easy to make a few improvements in this sentence because you have some great elements to work with. First, let's do a review of a few basic grammar and punctuation points, and then we can address ways to make the sentence flow better.
The first area we need talk about is the boundaries of a sentence. What you have right now is not one sentence, but two sentences. Notice that you have a period after "1968." Then you have a word that begins with a capital letter and a new subject and verb. So, that is a new sentence.
Second, you are using semi-colons in a way that does not help any of this to flow. A semi-colon is mostly used to join two independent clauses together. Here is an example for you:
I don't have much in the pantry that I can cook; I think we should go out for dinner.
Do you see how the clauses on both sides of the semi-colon could stand on their own as sentences? That is the most common use for the semi-colon.
Now, if you want this to be one sentence, you can do this without much difficulty simply with descriptive phrases and a compound predicate. Let's look at how to do each of these.
The subject of the sentence is your mother. So you should certainly begin with that. You can describe her in a descriptive phrase right after her name, placing the information about her birthplace and birthdate in that phrase, with commas to set the phrase off from the main parts of the sentence. Let me give you an example:
My grandfather William, born in Russia 1898, came to this country as a young child.
Do you see how I have placed that information in a phrase, set off with commas, and then have gone on in the sentence with more information? You can do this with more than one descriptive phrase, too. Here is another example:
My grandfather William, born in Russia in Moscow, a very large city, came to this country as a young child.
What I have done here is include two descriptive phrases, each set off with commas.
In order to provide the rest of the information you want in your sentence, we need to look at how to use the verbs you need to have a compound predicate. All you need to do is not repeat the subject,"mother," add a conjunction, and provide the rest of the information you need in the sentence. Here is an example of how I can use the example above to do this:
My grandfather William, born in Russia in Moscow, a very large city, came to this country as a young child and lived in New York City with his family.
Notice that I did not repeat the word "grandfather" in my sentence. This subject covers the rest of the sentence, too, just by adding "and" and the next verb.
You already have some good descriptive phrases in the second part of the sentence, with the names of your grandparents. All you need to do is eliminate the semi-colons in that section.
Often when we want to get lots of information in one sentence, we can do this with descriptive phrases. Making compound predicates instead repeating a subject often can help the flow, too.