What should I be mindful of in everyday mindfulness?

This is a good question.

Always the mindfulness should be in the 4 foundations of mindfulness. One objective of mindfulness is to remove defilements or letting go. Pahāna Sutta gives an outline in brief. Another is to keep you on the path as outlined in Maha Cattarīsaka Sutta. Any sensation is unsatisfactory this is the link to understand the 4 Noble Truths and also with sensation, according to dependent origination, arises craving. If you miss this link you miss the path.

That is the when taking the 1st foundation of mindfulness, the body, feel the elements like the temperature of the water, the texture of the plate, etc. and you reaction. The water is too hot, my hand is burning or water is too cold, etc., i.e., evaluation you give as this is desirable, undesirable or neutral and satisfaction, dissatisfaction and neutral feeling that follows. The look at the posture and position. The pain and resulting dissatisfaction from standing a long time near the sink, pleasure from changing postures when the previous posture becomes uncomfortable, the pain for wariness from hard work (movement - scrubbing), and the pleasure of rest, etc. [Dhātu Vibhaṅga Sutta, Titth’ayatana Sutta, Indriya Bhāvanā Sutta, Sal-āyatana Vibhanga Sutta, Dīgha,nakha Sutta]

The examin feelings in general. Are you evaluating something as favorably, unfavorably, or neutrally and the resulting feelings of pleasure, displeasure and neutral feelings. [Mahā Sati’patthāna Sutta, Dhātu Vibhaṅga Sutta, Titth’ayatana Sutta, Indriya Bhāvanā Sutta, Sal-āyatana Vibhanga Sutta, Dīgha,nakha Sutta, Cūla Vedalla Sutta, Mahā Vedalla Sutta, Samma Ditthi Sutta]

Also mental analysis of the situation. Plates are dirty hence displeasure. I have better things to do then displeasure. Or whatever else that might arise either pleasure, displeasure or neutral by your evaluation of liking, disliking or do not care. For more detail refer the 18 investigation in Dhātu Vibhaṅga Sutta, Titth’ayatana Sutta & Indriya Bhāvanā Sutta and also see Cha Chakka Sutta. Also note the feeling arising from whether you like it, hate it, what kind of thoughts are aroused (my friends are at a party while I am washing dishes) etc. and thinking pondering and planning that follows and what you feel about it.

Also see if you are transgressing any moral principles. There is an ant in the sink, I should not kill it and perhaps save it. [Maha Cattarīsaka Sutta]

Ultimately whatever feeling that arises you should be aware of its arising and passing nature.

Also see The Satipaṭṭhāna Suttas: An introduction and Vedanā (Feeling) by Piya Tan

In general situation, regarding the body, when you see a body you perceive as beautiful hence pleasurable and so on. Again in these situation you should be mindful of the perceptional evaluation beauty / smart, ugly, plain and the feelings that arise with it. [Maha Dukkha-k,khandha Sutta] You can also go to the extent of cemetery contemplations if you are very lustful to overcome Vipallasa. [Mahā Sati’patthāna Sutta, Kaya,gatā,sati Sutta, Maha Dukkha-k,khandha Sutta, Vipallasa Sutta]. Maha Dukkha-k,khandha Sutta more detail on the mechanics than the Mahā Sati’patthāna Sutta.

Also when wanting to answer a call of nature the feeling of tightness and the displeasure before and the subtle pleasure of release. [Mahā Sati’patthāna Sutta]

Also you can be mindful of the breathing. Is it deep or shallow. If you are displeased what happens and what happens when you are pleased and what is it like when you are neutral. [Mahā Sati’patthāna Sutta, Kaya,gatā,sati Sutta, Ānâpāna,sati Sutta]

Also the aggregates, the faculties and their experiences, phenomena, etc. all having the universal characteristics. Expecting something positive as permanent gives disatisfaction when it happens to be impermanent and likewise for other combination. [(Anicca) Cakkhu Sutta, Cūla Vedalla Sutta]. Also see: Nibbida by Piya Tan which has more ferences of relevant Suttas towards the end of the essay.

For example, when washing dishes do we focus on what we notice in our current experience;

Your current experience is what you sense or feel. So concentrate on what is felt like in the Pahāna Sutta. This is one of the simplest forms.

like as I wipe the dishes I can focus on the feel of my hands on the plate or the view or sound of the wiping.

Focus on both. Sounds may trigger other things. You may be irritated by the sound. Also it may trigger memories. Whatever sense door body (hand) or ear just be mindful of the evaluation and sensation. (18 evaluations above.)

Or do I decide to pick one thing(sense) and focus on that purpose for a few seconds?

In normal work no need to pick. Just keep a check that your awareness is present on some sensation or the other.

In simple terms, your focus should be on the arising and passing of sensations like in the Pahāna Sutta.

Or do I try to focus on everything in the now which seems like information overload?

Your focus should be on what is felt (feelings / sensation) and their impermanence as in the Pahāna Sutta.