Haywards Heath is on the main Brighton line, with frequent Thameslink services from St Pancras, Blackfriars and London Bridge, plus Southern trains from Victoria. The journey time from London Bridge or Victoria is around 40-45 minutes. On the Thameslink route you can get cheap Super Off-Peak tickets from central and north London (Sat, Sun & BH), but these are not valid on Southern services.
Lindfield is served by several bus routes from Haywards Heath, but only one operates on Sundays and Bank Holidays. Metrobus 270 runs hourly (two-hourly Sun & BH) and Metrobus 272 two-hourly (Mon–Sat). There is also an hourly local town service, Compass 30 (Mon–Sat). The last service back from Lindfield is at around 6pm (5pm Sun & BH).
If you are planning to take the bus in both directions you can save money by asking for PlusBus when you buy your rail ticket to Haywards Heath. With a railcard, the additional cost is about the same as one single bus fare.
If driving, Haywards Heath station car park costs £9.20 Mon–Fri, £5.30 Sat, £2.25 Sun & BH (2020). Car drivers might prefer to do the Short Circular Walk as there is a free car park in Lindfield, signposted from the High Street by the Bent Arms.
The only bus service from Sheffield Park Station is on Saturdays, when Compass 121 runs two-hourly to Lewes (last bus 5pm; note that it does not pick up passengers from the NT Garden). In theory you could change at North Chailey for Compass 31 to Haywards Heath, but the connection does not work well.
As noted above, on the Bluebell Railway Walk you travel via East Grinstead from Victoria (half-hourly). To complete the journey you need to buy an All Line Return ticket on the heritage railway (£19 in 2019, cheaper if bought in advance). This covers the journey back from Sheffield Park and admission to all the station facilities (locomotive shed, carriage works, etc).