The considered guideLondon
London: two ways to make an art day
Bankside for modern art, Trafalgar Square for painting, and a Covent Garden table when the day calls for dinner.

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Choose a gallery before choosing a route. Tate Modern gives you modern and contemporary work beside the river; the National Gallery offers a different starting point around Trafalgar Square. Both have free collection entry, while special exhibitions need their own admission check.
Rules can be a separate meal in Covent Garden, with a reservation and time to get there. There is no prize for doing both museums in one day. Pick the work you most want to see and let the rest of the afternoon take shape around it.
Tate Modern
Choose the Bankside version for the modern collection or a specific temporary show. A riverside pause after the galleries makes a natural change of pace.
National Gallery
The Trafalgar Square alternative. Pick a few paintings to spend time with rather than adding a second full museum to the day.
Rules
A table for the evening chapter, especially if you want a traditional British menu. Reserve and check service cutoffs before fixing the rest of the day.

