North/South/East towards anywhere in England
The key to getting to pretty much anywhere in England from Chester is to get to the, via the . You should be able to get to the M6 in one lift with a sign saying "M6 North/ South", as lots of people will be heading in that direction (anyone going to Manchester/ Birmingham/ London/ Scotland etc.).
Even if they are going in the wrong direction after that there is a service station (Lymm services) where the M56 meets the M6 where they should be able to drop you off. Note - this service station is NOT directly on the motorway, but a little bit off it round a roundabout - so pay attention to road signs! It is at junction 9 of the M56 and junction 20 of the M6. If they are going south, they could also drop you off at the Knutsford services at junction 19 on the M6.
To get onto the M56 in the direction of the M6 is not too difficult at all (JackW has done it quite a few times, and almost always waits less than 30 minutes). Near the train station there is a road bridge over the train line - this is Hoole Road. Walk over the bridge along this road and just keep following the road out of town. After about twenty minutes you will get to a roundabout. Cross over to the second exit, which leads onto the M53 (lots of people go this way onto the M56 and then onto the M6). On your left, there is a small residential road running parallel to the main road (called Greenfield Crescent - see https://goo.gl/maps/HtdJY). At the end of this road, it joins the main road, creating a space for cars to stop in - this is your spot!
West towards North Wales
Chester is on a lot of routes towards the North coast of Wales. The A55 is the road you want here - it takes you all along the coast to Anglesey (or you can leave it at Bangor to get to places like Abersoch, on the Llyn peninsula). Once you get onto the A55 you should have no problem: It is a beautiful road and there are lots of lay-bys, and even some burger vans if you're lucky. The hardest bit is actually getting onto in the first place.
The junction you use to access the M53, when heading towards Manchester (see above) is also somewhere you can get onto the A55 heading in the other direction. The problem is that the convenient place you can stand to get onto the M53 is too close to the junction, so the drivers going to the A55 are already in the inside lane, making it less likely for them to stop. This has worked for JackW in the past, but expect to wait a bit longer than if you are going towards the M6
Probably a better option is to head out of Chester to the south. You can either get the number 1 bus (from the train station to the Business Park Holiday Inn, about 15 mins) or walk (about 45 mins - 1 hr). The walking route is to walk over the river on the A483, past the racecourse, and take the second road exit (not including the pathway) from the roundabout you come to, called Wrexham Road. Then walk down past the two roundabouts that go into the business park until you find a suitable place to stand - this is near where cars get onto the A55.
If you get the bus, this Holiday Inn is near the roundabout where cars get onto the A55 and there should be several places you can wait where cars can pull in.
If you get dropped off at Lymm Services (Jn 20, M6) or Knutsford Services (Jn 19, M6), it is normally possible to get to Chester in one lift with a sign that simply says "Chester".