Great question — May is actually one of the better months to road trip from Delhi if you pick the right destination. Here's what I'd recommend based on personal experience:
Shimla is my top pick. Roads are smooth all the way, temperatures stay around 15–20°C and it doesn't get unbearably crowded until late May. The Kalka-Shimla highway is in great shape right now. Most groups I know book a
tempo traveller in Delhi for this route — one vehicle, no coordination headache, door to door.
Mussoorie is another solid choice — the Delhi-Dehradun expressway has made the drive genuinely enjoyable, under 5 hours from most parts of Delhi, and the hill station sits comfortably at 18–22°C through most of May.
Lansdowne is the one most people sleep on. Only 250 km from Delhi, roads are well maintained, cool weather throughout May and you'll barely run into tourist crowds. Highly recommend if your group wants a relaxed trip over a busy one.
Nainital works but go before mid-May. Once school holidays kick in around the third week it gets genuinely chaotic. Roads via Moradabad are fine and the lake town stays cool and pleasant in early May.
Rishikesh if you don't mind slightly warmer weather — around 28–30°C — the roads are among the best on any Delhi outstation route and the vibe in May before monsoon is hard to beat.
Stay away from Leh-Ladakh and Spiti in early May. The highways are often still closed or risky. Wait until June at the earliest.
Hope this helps — which direction are you leaning?