...I am on a journey finding out about my chosen era....there is a lot more to it than just buying coaches and plonking them on the track...
If you are aiming at reasonable fidelity, coaches are a major challenge. The LMS started - inevitably - with no coaches of its own design; and even by the time BR came along 25 years later was still operating a great many pre-group design coaches, in addition to its own extensive build of coaches in three distinct styles (usually referred to as Period 1, 2 and 3, a classification devised by certain well known LMS modellers studying its operations, which BR then adopted to aid in its understanding of the vintages of the inherited LMS constructed coach stock!).
Some careful choice of location and date span might aid in keeping the required 'coach motley' under control - I don't know enough about the LMS to advise on the specifics. (As an example I have managed this aspect on my BR(ER) steam operation, by choosing a start date by which the large majority of the inherited pre-group stock had been swept away by BR, as it's large scale mk1 vehicles build had by then enabled their removal from revenue service for scrapping.) And because the LMS had direct connections to all the other UK railways, 'through coaches' from the other groups may or may not be up for consideration, your choice...
A book I often refer to, and which is usually obtainable s/h, is 'The Big Four in Colour' the contents of which are exactly as the title suggests.
Wagons now. The LMS had the largest wagon fleet of the four groups, near half, followed by the LNER with well over a third, these two operating over 80% of all company wagons, the two small groups contributed the rest. With a large proportion of these company wagons 'common user', they were mixed up all over the network.
A book I often refer to, and which is usually obtainable s/h, is 'The Big Four in Colour' the contents of which are exactly as the title suggests.