We don't know where young Richard was at the time of his father's execution, although the most likely place would be with his stepmother, Maud Clifford, up at Conisbrough Castle.
In October 1417 he passed into the care of Sir Robert Waterton, a Yorkshire knight with a long record of service to the House of Lancaster, and previous experience in looking after spare royal children. However in 1423 his wardship and marriage were purchased by Ralph Neville, Earl of Westmorland.
Neville was another long-standing supporter of the Lancastrian monarchy, defecting to Bolingbroke ahead of the crowd in 1399 despite lavish favour shown him by Richard II. His (second) wife was of course the famous Joan Beaufort, half-sister of Henry IV, and their hobby was arranging impressive marriages for their children. By October 1424 Richard was already betrothed to their youngest daughter, Cecily. Cecily was accounted a great beauty in later years, and may have inherited this from her grandmother, Katherine Sywnford.
Ralph Neville died a year later, but that did not prevent a marriage between Cecily and Richard in 1429. Cecily was 14 on 3 May 1429, Richard about 18. It's possible the marriage was consummated at this time, but no children were born for ten years. As the couple were definitely fertile it may be they spent little time together in the early part of their marriage.
Richard was knighted by the Duke of Bedford in 1426, and his next public outing was at Henry VI's coronation in late 1429.
The most useful textbook for Richard Duke of York is Duke Richard of York 1411-1460 by P.A. Johnson. This is highly recommended and particularly useful if you want the full SP on Richard's complex financial affairs. (They are far too complex to be covered in a blog, but also very interesting.)