Why would you allocate a vector with new? The main point of using vectors is to avoid having to use new.
vector<int> v;
v.push_back(1);
cout << v[0] << endl;
If for some strange reason you decide you really must use a pointer, then you can do
vector<int>* v = new vector<int>;
v->push_back(1);
cout << (*v)[0] << endl;
But really, allocating a vector with new makes little sense.
Perhaps you were a Java programmer before you tried C++? If so then don't try to program C++ in a Java style, they are very different languages. You will get into a horrible mess if you do.