Oudeschild itself is tiny — everything is walkable in ten minutes. The harbour, Museum Kaap Skil, and the village centre are all within easy strolling distance of each other.
For getting around the rest of Texel from Oudeschild, a bicycle is by far the best option. Texel has one of the best-maintained cycling networks in the Netherlands: 140 km of dedicated fietspad (cycling paths) circle the island and cut across it, running through dunes, polders, and nature reserves. Bikes can be hired in Den Burg and at several other points on the island. A day's rental gives access to all of Texel's villages and natural areas.
The Connexxion bus network covers the island adequately, with Den Burg as the hub. Several routes pass through or near Oudeschild. A day ticket covers unlimited travel on all routes.
Cars are unnecessary for village visits and actively awkward — parking is limited and cycling or walking is more enjoyable in almost every situation.