Hampton Court Palace is huge, and there are several routes through different parts of the Palace. It is worth considering which era means the most to you and do that first so you aren't tired out for your favourite. By early October the gardens, especially the Borders, won't be at their best, but the Rose Garden should still have some nice flowers if the frost hasn't come yet, and the Privy Garden looks fabulous in every season. The Great Vine - the world record holder - is worth seeing although all the fruit and much of the leaves will be gone, and the nearby walls of colourful autumn leaf colour should be bright red. Around from the Great Vine is the Banqueting House - enquire if it will be open - is very different and has neat history and exotic paintings, and the walk from there to the Privy Garden gives lovely views of the Thames.
On the opposite side of the Palace is the world famous maze. Plan on getting lost and taking up to half an hour to find the centre.
I am particularly enamoured of the Great Hall, the Chapel, the Clock, the Fountains and the audio background in the Court.
The restaurant outside between the vegetable garden and the rose garden has excellent food, very well priced, and very healthy. (You don't have to be a child to have Bangers and Mash although the sign says you do). The fill-you-own-plate salads taste as fabulous as they look. Free chilled tap water is available from the condiment stand. If you are near the chapel and Royal Kitchens there is a very small snack bar - I've gone in and have never stayed to get anything, I much prefer the large one I've just described. I've also never been in the fancy restaurant near the Chocolate display.
If you happen to be HRP members (you may be able to guess that I am) you get 10 percent off in the gift shops and the restaurants.
There is no way on God's Green Earth that you will see all of Hampton Court Palace on a first one day visit. Go online and study their website and plan what you want to see and the routes you want to take.
By the way - you will never have any problem finding a place to sit down, nor will you be lacking for a toilet. There are lots of both. Look around in the toilets, they are in genuine parts of the Palace and some of the architecture is interesting in there too.