There are about 57 species of butterfly in the UK, but over a whopping 2500 moths! Although most moths are nocturnal, you might see some day-flying ones out and about.
If you do want to see nocturnal moths you can make a trap in your garden: https://butterfly-conservation.org/in-your-area/east-scotland-branch/budget-bucket-moth-trap, or even just hang up a sheet with a light behind it: https://www.lancswt.org.uk/blog/charlotte-varela/moth-trapping-beginners
Here are a few common, easy-to-spot butterfly and day-flying moth species
Peacock butterfly Aglais io
Photo: Uoaei1 CC BY-SA 4.0, Wikimedia
Red Admiral butterfly Vanessa atalanta
Photo: Evelyn Simak CC BY-SA 2.0, Wikimedia
Small tortoiseshell butterfly Aglais urticae
Photo: Kenneth Allen CC BY-SA 2.0, Wikimedia
Large white butterfly Pieris brassicae
Photo: Uoaei1 CC BY-SA 4.0, Wikimedia
Orange-tip butterfly Anthocharis cardamines
Photo: Charles J. Sharp CC BY-SA 4.0, Wikimedia
Painted lady butterfly Vanessa cardui
Photo: Anna N Chapman CC0 1.0, Wikimedia
Cinnabar moth Tyria jacobaeae
Photo: Charles J. Sharp CC BY-SA 4.0, Wikimedia
Emperor moth Saturnia pavonia
Photo: Ben Sale CC BY 2.0, Wikimedia
Currant clearwing moth Synanthedon tipuliformispavonia
Photo: Whitney Cranshaw CC BY 3.0 US, Wikimedia
These are useful online resources for identification
Butterfly Conservation butterfly guide
https://butterfly-conservation.org/butterflies/identify-a-butterfly
Butterfly Conservation moth guide
https://butterfly-conservation.org/moths/identify-a-moth