The VAT aspect is straight forward - not paid on export paid on import. For example I buy from , say, Rimmers, at their Ex VAT price then pay French VAT on arrival here. No import taxes as such.
The complications come with things such as cars assembled in the UK coming to the EU. If over 55% of the value is of UK origin then no tarif to pay. If it is less than 55% then there are import tarifs to pay. For the moment the deal is giving the UK 18 months, I think, to increase UK value content. With cars it is currently around 35/40%. To be honest I don't know if EU produced parts going into a UK assembled car count or not.
All very complex, no winners only losers.
Last week travelers arriving at the Hook of Holland from the UK had their sandwiches confiscated. The reason is that meat and dairy products are strictly controlled now that the UK is outside the EU as they are for other countries. M&S food shops in France, there are a few, have been out of stock of their famous sandwiches for this reason.