If the population you want to study is healthy older adults, this is a great place to start. This comprehensive MRC report provides an overview of key cognitive abilities, and the tests used to assess them. It's also particularly useful for describing some of the practicalities to consider if you are developing a cognitive assessment.
There are a number of tests that are particularly sensitive to the changes that occur in preclinical dementia - the clinically silent phase preceding dementia diagnosis. If you're studying the early stages of dementia be sure to check out these recommendations.
Quick cognitive screening tests such as the MMSE, MoCA and ACE-III can be useful, though relatively crude, indicators of whether participants have a cognitive impairment. Screening tests and more detailed cognitive tests are both useful tools but it’s important to be aware of the differences.