If you're staying in the city center (Puerta del Sol, Gran Vía, Plaza Mayor), enjoy a direct walk to the Royal Palace of Madrid. It’s a 10–15 minute walk through pedestrian areas and historic sights.
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Lines 2 and 5, with a stop at Ópera, drop you just steps from the Royal Palace of Madrid. One exit from the station places you in Plaza de Oriente.
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Lines 3, 25, 39, and 148 stop near Plaza de Oriente, right in front of the Royal Palace of Madrid. From there, it's just a few minutes' walk to the Esplanade of the Almudena.
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Take the train to Príncipe Pío (lines C‑1, C‑7, C‑10). From there, a pleasant 10–12 minute walk through gardens and historic streets leads you to the Palace entrance.