那我们快去医院吧。    那：used  as  conjunction.  means :then,  in  that  case.

我见到妈妈真高兴。    到：used  as  the  verb  见's complemet.

hope  u  can understand ,  maybe  it's  not  so  clear.

thank you so much ! now it's really better ! :) 

那 functions much like "Then" at the beginning of English sentences, offering a suggestion as to what to do next.

我听说妈妈刚刚住院了。 I heard mom was just hospitalized.

那我们快去医院吧！ Then let's get to the hospital, quick!

到 at it's most fundamental level is to arrive at something.

我昨天到医院找妈妈，但找不到她。 I went to the hospital yesterday looking for (my) mom, but I couldn't find her. 找不到 literally is "find+not+arrive," so it implies that you went searching for her, but you were unsuccessful.

Another example:

我看不到电视，因为有人挡我前面。 I can't see the television, because somebody's in my way.

I look+not+arrive = You're trying to look at it, but you were unsuccessful.

1.) “那” at the beginning of a sentence is pretty informal.  It's like saying.."hmmm well, we should quickly go to the hospital",  or, "sooo we'll go quickly to the hospital, yeah?"

I can't really think of an "accurate" translation but that's the sense of it (and of course it can be translated differently depending on context).  I think the best way to understand how to use it is just to see how native speakers use it.  

2.) you should just take "见到“ as a whole "two character" verb to mean  "to meet".  There's little point in dissecting it because it doesn't make much sense translated.  You'll find there are HEAPS of "two character" verbs which contain "到“， eg: 找到 (to find)，遇到(to encounter)，得到(to obtain/get) and on and on.  None of these verbs have anything to do with "arriving" which is what 到 normally means on its own.

加油！:)

here "那"  means  " then",    i.e.  "  allora " in italian .

here "见到"  emphasizes the result ——you have seen your mather,and this makes you quite happy!