How can you speed up or improve your metabolism?
You can influence your metabolism, but the effects are more modest than popular claims suggest. Most of the gain lies in more everyday movement, more protein in your diet, and regular exercise, not in special supplements or meal-frequency tricks.
Everyday movement that you barely think of as exercise counts for a great deal. Walking to the supermarket, standing while working, gardening, or simply fidgeting: the sum of all these small activities largely determines how much energy you burn each day. Your body even adjusts this automatically: eat more, and such spontaneous movement increases; eat less, and it drops on its own. It is therefore one of the most powerful yet most underestimated dials you can turn.
What you eat also determines how hard your body has to work to digest it. Protein and carbohydrates cost the body more energy to process than fat does. Larger meals produce a greater thermic effect than many small snacks spread throughout the day, contrary to what you often hear. A low-fat, plant-based diet appears to increase the thermic effect of meals. The popular claim that low-carbohydrate or ketogenic diets give the metabolism an extra boost is, incidentally, incorrect: when calorie intake is the same, a low-fat diet produces on average 26 kilocalories per day more energy expenditure and 16 grams per day more fat loss than a low-carbohydrate diet. Small, but it works in the opposite direction from what many people think.
Regular aerobic exercise and strength training also help counter the metabolic slowdown that comes with ageing. Hormone replacement therapy is discouraged in most cases because the risks outweigh the benefits; exercise and modest caloric restraint are the recommended route. Exercise also stimulates the production of the hormone irisin, which in mouse studies converted fat storage into fat burning. Whether this works equally strongly in humans is not yet clear.
If you want to lose weight, be aware of an inconvenient countereffect: the body slows its metabolism further when you lose weight, beyond what you would expect based on your lower body weight alone. This makes sustained weight loss harder. Well-designed studies, however, found this effect to be smaller or barely present, and after a period of stable weight it appears to diminish. Physical activity is the best-documented factor for maintaining weight loss over time.
NAD supplements such as nicotinamide riboside are sold with the promise that they improve metabolism. In mouse studies they increased energy production in cells and protected against the effects of a high-fat diet. But all results come from animal and cell research; in humans this effect has not yet been demonstrated. Do not buy them on the basis of those mouse results just yet.
This answer draws on multiple reviews and controlled dietary studies, including a systematic summary of 32 controlled dietary studies on diet composition and a systematic review on adaptive thermogenesis. The findings on NEAT, dietary composition, exercise in ageing, and diet type rest on reasonably consistent evidence in humans. The irisin research and NAD supplements are largely based on animal research; the researchers themselves note the limited translation to humans. For adaptive thermogenesis, the researchers rated the quality of most studies used as moderate, and well-designed studies found smaller effects.