Salford is not a suburb of Manchester, it's a city in it's own right
As someone who lived in Greater Manchester for 7 years and explored the county extensively, I would say Salford actually does feel like a distinct city rather than a mere 'Suburb of Manchester' in some places (especially around Pendleton Precinct and Salford Crescent) given it has its own university, cathedral, museum and commercial centre.
On the other hand, if we're using official city status, this could also open up the possibility of 'City of London' and 'Westminster' within Central London being counted as two separate cities, which has countless rail routes between them, including Paddington (Westminster) to Liverpool Street (City) on the Elizabeth Line, Bank (City) to Westminster on several tube lines, the list goes on!
That's why I put "(arguably)" England's second city. "As the second city is an unofficial title and one of subjective opinion, a number of polls have been conducted over the years. Despite Birmingham being viewed as the traditional second city, public polls have shown a slight preference for Manchester since 2000. "
en.wikipedia.org
I see both Manchester and Birmingham as "Joint 2nd" as their overall urban populations are neck-and-neck and I consider them equally good in different ways. For example, Birmingham is a greater hub for domestic transport (better variety of destinations across the country by both coach and train) while Manchester has the edge internationally (its airport is much bigger and busier with a better variety of destinations).
BACK TO THE TOPIC:
- Manchester to Liverpool is a very prominent one, oldest link in-fact between two cities in the world, with the current connections being either the (still) unelectrified CLC line via Warrington Central and the electrified Chat Moss line via Newton-le-Willows; most of the latter was part of the original 1830 line back when Stephenson's Rocket ran on it.
- Manchester to Blackburn has a direct line via Bolton and a longer-way-round via the Todmorden chord; both run opposite directions out of Victoria, forming a nice loop.
- Liverpool to Preston has a through service running via Wigan and another line where you have to change at Ormskirk due to how separate Merseyrail and Northern are kept.
- Liverpool to Chester has the Merseyrail service via the Wirral and now a TfW service running via the Halton Curve.
- London to Southend has the Great Eastern to Victoria (GA) and the LTS to Central and Shoeburyness (c2c).
- Manchester to Leeds has the direct Huddersfield line (TPE) and the Caldervale Line (NT) which itself is split via Halifax/Bradford and Brighouse.
- Edinburgh to Glenrothes can be done either to Glenrothes with Thornton via Fife Circle or SECML to Markinch, which is actually closer to Glenrothes town centre than its namesake station.
- London to Exeter has three routes: two GWR routes, one via Bristol and one (two depending on how you look at it) via Newbury (with a loop via Westbury); and the slower SWR route via Salisbury.
- London to Southampton also has three: direct via Winchester, PDL via Haslemere and Southern via Horsham.
- Crewe to Stafford has both a direct line without stops and a line via Stoke-on-Trent.