Not served by the Skytrain (BTS), your options for getting around Rattanakosin are limited to taxis tuk-tuks, walking or river taxis. We strongly recommend the latter. It's cheap, photogenic, and the stops dotted all along the Chao Phraya River hold many surprises in store. See our Chao Phraya River Guide.
Not imposingly large or hard to undertake, it's also a fascinating area to walk around, provided you have a comprehensive map in hand. Alternatively, with short distances between most sights, the Old City is a good place to commandeer a tuk-tuk, provided you agree a price before setting off.