You procedure has 2 major problems. In order to return values a procedure must include an OUT or INOUT parameter. Secondly a Select inside a plpgsql block must include the into clause. Further this parameter must match number of columns and data type of each column. For what you have this means creating a user defined data type (UDT). The results then becomes: (Assumed a definition of table routes since it was not provided.)
create type routes_category as (flight_num     integer
                               ,distance_miles integer
                               ,from_city      text
                               ,to_city        text
                               ,flight_time    interval
                               ,category       text
                               );
create or replace procedure distances(in  flight_num1 integer
                                     ,out flight_dtl  routes_category
                                     )
language 'plpgsql'
as $body$
begin
    select 
        *,
        case
            when distance_miles >= 0 and distance_miles <= 2000    then 'short distance travel (SDT)'
            when distance_miles >= 2000 and distance_miles <= 6500 then 'Intermediate distance travel (IDT)'
            when distance_miles > 6500                             then 'Long distance travel (LDT)'
        end as category
    into  flight_dtl 
    from
        routes 
    where 
        flight_num = flight_num1;
end;
$body$; 
But why are you using a Procedure at all, it does nothing but retrieve data. If this is actually the case you should use a function. You still need to define the appropriate return type, either as above or as the return type as table. If it contains just SQL then define it as a SQL function.
create or replace function distances_fun(in  flight_num1 integer)
  returns  table(flight_num     integer
                ,distance_miles integer
                ,from_city      text
                ,to_city        text
                ,flight_time    interval
                ,category       text
                )
  language sql
as $body$
    select 
        *,
        case
            when distance_miles >= 0 and distance_miles <= 2000    then 'short distance travel (SDT)'
            when distance_miles >= 2000 and distance_miles <= 6500 then 'Intermediate distance travel (IDT)'
            when distance_miles > 6500                             then 'Long distance travel (LDT)'
        end as category
    from
        routes 
    where 
        flight_num = flight_num1;
$body$;
Demo for each here.
Unfortunately, db<>fiddle does not show results from raise notice .... That would have been:
Flight 1111 is short distance travel (SDT) 
Flight 1122 is short distance travel (SDT)