The closest airport to Uddevalla is Trollhätten-Vänersborg airport, which lies about 30 minutes away by car. The easiest way to get there is by airport taxi. The airline Golden Air flies regularly to Bromma. The flight time to Stockholm is just under an hour.
You can, of course, also fly via Gothenburg’s Landvetter airport or with Ryan Air which uses Göteborg City Airport at Säve. There are regular bus services from these airports into central Gothenburg, from where you can take a bus to Uddevalla.