list1={1,2,3,4,5};
list2={10,20,30,40,50};

list3={1+10,1+20,1+30,1+40,1+50,2+10,2+20,2+30,2+40,2+50,....,5+10,5+20,5+30,5+40,5+50}

Total[Tuples[{list1, list2}], {2}] (* or *)
Join @@ Outer[Plus, list1, list2]

Tuples[Inactive[Plus][list1, list2]] (* or *)
 Join @@ Outer[Inactive[Plus], list1, list2]

Plus @@@ Tuples[{list1, list2}]

Flatten @ Outer[Plus, list1, list2]

Flatten[Table[
  ToString[list1[[i]]] <> "+" <> ToString[list2[[j]]], {i,
   Length[list1]}, {j, Length[list2]}]]