Butter can easily be replaced by soya/sunflower spread. If you're intentionally making a vegan cake, make sure that these are really vegan; quite often sunflower spread can have added milk.
To check whether a cake is done cooking or not, stick a cocktail stick/skewer/sharp knife into the centre. If it comes out clean, the cake is ready to come out of the oven, if not, give it five more minutes and check again.
Eggs are generally the most difficult thing to relace when making non-vegan cakes vegan, as they're needed to bind the mixture. However, interestingly, some cake recipes just happen not to contain eggs, especially if they date back to war time. You can buy egg replacer, which works well, though I do always feel like that's cheating. You can also use banana, flaxseed, tofu, soya yoghurt or vegetable oil. Though you should be careful with the last one, as sometimes vegan cake recipes can be a little over-oily; to counter this, just add less oil. For more information, see what the Post Punk Kitchen have to say about replacing eggs.
If you don't have self raising flour, add baking powder to plain flour (1 tsp for every 75g).
Milk can be replaced by soya milk, rice milk, oat milk or any other kind of non-dairy milk. There are all kinds of weird and wonderful things out there, like quinoa and almond milk.
1 table spoon (tbsp) = 3 tea spoons (tsp)
Don't use olive oil in cakes; the flavour is too strong. I know this from disappointing-cake experience.