There are a vast range of possibilities and it will not be possible to carry out everything at once so some will have to wait and take their turn. It's hard to know where to start but in no particular order here are a few examples:
Biochemists might want to look for life in water aquifers that are thought to exist deep underground. This could involve multiple samples from multiple sites and multiple depths with great efforts to prevent contamination. Materials extracted could be examined for the presence of DNA and other compounds that might be indicative of life such as amino acids and complex organic molecules. Analysis might use a DNA sequencer and a mass spectrophotometer. As with a lot of research it can be very dull and repetitive with hundreds of samples requiring the same treatment.
Paleontologists might want to look for micro fossils in a wide range of locations. This might be as simple as using a geologists hammer to break open likely looking rocks and a microscope to study them.
Geologists might be interested in characterizing and dating various features in the Martian terrain. This could involve samples from a range of areas as well as core samples from a range of depths. Analysis might involve a powerful microscope to look at crystal structure and a range of instruments such as x-ray diffraction and x-ray fluorescence spectrophotometers to help identify the chemical composition. Note a range of radio-dating techniques are available other than those based on carbon
Materials scientists and technologists might be interested in developing manufacturing processes. For example making bricks out of regolith for use in construction of habitats, bunds or for radiation shielding. What type of regolith, particle size distribution and what additives might best be used for making the strongest bricks? And how much pressure is required to form the bricks? They might need a press for brick manufacture and testing, sieves and a range of additives.
Others might be interested improving the solar cell efficiency or learning how best to extract nickel or iron from regolith perhaps using the MOND process or how to better detect and remove perchlorate contamination.
Biologists might be interested in studying how best to maintain and improve the biosphere that provides the water, oxygen and food that a base uses.