Crossrail's a bit of a red herring as far as Kingston's transport
needs go. There are already two perfectly good railway lines to London
and no-one would want to get onto the District/NLL or any line running
through over it at Richmond as it's so bloody slow. The directions
poorly served are on the other side of the town. What I'd recommend
would be a tramway network with the following lines:
1) to Epsom via Surbiton, Tolworth, and Ewell Village (running on Wood
St, Eden St, Brook St, Penrhyn Rd, Surbiton Rd, Surbiton Hill Rd,
Ewell Rd, Tolworth Bway, Kingston Rd, High St Ewell, Epsom Rd, East
St, High St Epsom, Waterloo Rd, Station Appr)
2) to Leatherhead via Surbiton, Chessington, and Chessington WOA
(branching off (1), from Ewell Rd running on Ditton Rd, Hook Rd,
Leatherhead Rd, ROW parallel to Kingston Rd, alongside the railway,
Station Appr, Station Rd, North St, High St Leatherhead)
3) to Heathrow T4 via Hampton Court, Hampton, and Feltham (running on
Wood St, Clarence St, Kingston Bridge, Hampton Ct Rd, Church St, High
St Hampton, Uxbridge Rd, Hampton Rd W, Uxbridge Rd, Hanworth Rd,
Hounslow Rd, Bedfont La, Staines Rd, Gt SW Rd, then into the Airport)
4) to Weybridge via Hampton Court, East Molesey, and Walton-on-Thames
(branching off (3), from Hampton Ct Rd running on Hampton Ct Bridge,
Bridge Rd, Walton Rd, Terrace Rd, High St Walton, Ashley Rd, Queen's
Rd, Hanger Hill, Station Appr)

I love it. Trams past my front door again - took me back to trolleybuses
past the door in the house I grew up in. However, some of those routes are a
trifle optimistic. I don't quite see how you'd make some of the curves on
those routes (very short trams needed!) and some roads like Ewell High
Street would present a challenge - but you get my vote!

Thanks! I guess by optimistic, you're suggesting my American-style
Interurban to Leatherhead is a bit much!

As far as the route to Epsom goes, the section as far as where the
Ewell By-Pass branches off the Kingston Rd isn't a problem - the trams
could even have their own lanes down the middle of the A240 dual
carriageway (that REALLY would bring a smile to my face in rush hour)!
With Ewell High St, the really problematic section would be from the
Chessington Rd to the sharp right outside the Star (where that awful
mini-roundabout is). Ideally I'd like to make this bit trams and
pedestrians only (it would certainly make crossing the road a lot less
lethal), maybe with a bit of gauntletted/single track to deal with the
curves.