Java (OpenJDK 8), 70 bytes

a->{int i=1;int[]c=new int[a.length+i];for(int u:a)c[i++]=u;return c;}

Try it online!

Explanation:

int i=1;    //Initialize counter
int[]c=new int[a.length+i];    //declare new array with length: 1 + input array length
for(int u:a)    //Loop through each element in input array
  c[i++]=u;    //Increment counter after shifting each element up one spot in the output array
return c;    //return new array starting at index 1