Where to go depends on whether you are going in early June or the end of the month
If in early June, I would recommend the Latterbarrow Cumbria WT Reserve just off the Barrow-in-Furness road on the opposite side to the turning for Ulpha and near a recently renovated white pub. The access road used to be the main road before the dual carriageway was built and there is ample (shaded) parking at the end as it loops back towards the main road (dead end).
This site has some fine Lancastrian whitebeams and also a strong colony of northern brown argus (which tend to fly early here, from about the 10th June and can be over by early July in hot years). It's quite a small site so it can be explored comfortably in an hour or two. The calcareous grassland is really spectacular and has columbine and lots of greater butterfly orchids
Just across the road is Meathop Moss where there is cranberry right beside the boardwalk. Take midge repellant if your'e visiting this site!
Humphrey head has hoary rock rose on the long slope dipping gently seawards at the end on the west side. It's tucked away within a much larger colony of common rock rose but if you follow the fence line from the footpath and find a double-post, you won't be too far away.
Much later in the season, spiked speedwell is on the western cliff, visible at distance from the path around the base of the cliffs and just after a memorial stone to a boy who died there in the 19th Century. This is at its best in early August
Unfortunately, many of the rarer species will not be in flower in June although I can give you sites for touch-me-not balsam and alpine enchanter's nightshade for future reference as well as a site for water lobelia that does not involve a long hard uphill slog! All these plants will be out in July so if your visit is at the end of June you may just get lucky.
All of these species are mentioned in publicly available literature but if you want any further information, feel free to contact me. I will be away doing field work at times so please bear with me if there is a delay in replying.