Design of circuit and soldering onto veroboard
That circuit is reasonably complex as it has discrete components, ICs, transistors and relays. My general approach would be

Reserve a track at the bottom for GND.
Reserve two tracks at the top for the +8V and +15V busses.
You can draw a coloured lines across the top of the board to mark the busses. e.g. Black for 0V, red for 15V and blue for 8V. It can be a good idea to mark every fifth line on top and bottom so that you have some references when you invert the board.
Lay out components in general with negative terminals towards the bottom.
Can any components be connected directly to the negative or positive rails? In your case all three transistors' emitters can be connected to the ground rail so they can be mounted at the bottom edge of the board.
Look for any components that can be aligned so that they can be connected as required without cutting traces.
When you run out of easy options then you have to start cutting traces and adding jumper wires. When adding jumpers you have two design choices:

Run wires from point to point.
Only allow vertical links (as shown in your diagram) and use tracks for horizontal links.




Figure 1. A tidy layout. Source: PCB Design SDL.
In the example above we can see that the designer has succeeded in keeping all the links perpendicular to the tracks. Notice all the track cuts under the LEDs, IC1, R1, R3, etc. See my answer to this question for more ideas.
Take your pick from online+stripboard+designer and start laying out.