Please clarify. What kind of route do you want? Do you want (a) the user to manually draw a route on the map? (b) the developer of the application/web site to draw a particular route on the map? (c) the user of the web site to determine a start and destination and then a routing engine find the roads to travel to get from A to B?
You can use OSRM as a back-end routing engine. If OSRM doesn't fit your needs (although seems as though it does) you can check out other routing engines on the Wiki page.
Please clarify. What kind of route do you want? Do you want (a) the user to manually draw a route on the map? (b) the developer of the application/web site to draw a particular route on the map? (c) the user of the web site to determine a start and destination and then a routing engine find the roads to travel to get from A to B?
(c) the user of the web site to determine a start and destination and then a routing engine find the roads to travel to get from A to B