Old Lady traveling alone- Gatwick to Cambridge

Take the Thameslink, towards Bedford, to St Pancras International. Get out of the station(which is on the lower level) and walk to the upper level. Once you are on the upper floor you can see signs to follow towards King's Cross Station. If in doubt ask any of the personnel present, they will be happy to guide you. From King's Cross station you can get a direct train to Cambridge.

In the main hall at King's cross there are display boards to guide you which train to take to Cambridge.

Map of St Pancras

UPDATE: As of now you can get a direct train to Cambridge from Gatwick.

As from now, there is a direct rail service from Gatwick Airport to Cambridge station. At the moment (March 2018) that is one train per day at 12:03 pm, but it will be extended to eventually four trains an hour soon.

These are Thameslink trains, and will be both the fastest and most convenient way to make this journey.

Just to add to @dumbcoder's answer sicen you asked for specific advice

St Pancras International is on three levels. When you arrive on the Thameslink you are on the lowest, below ground level. Take the escalator or lift to the exit. Keep your ticket handy to operate the gates. There is a wide gate if you have a wheelie bag with you, I think it is to the left. When you exit the gates turn left and you will see the exit you need on the far side of the concourse.

When you exit STP you cross the road and you are in King's Cross station. The departure boards are in front of you. Beyond them are the low-numbered platforms from which the long-distance trains depart. To your left is a smaller block of platforms from which, I my experience, the trains to Cambridge usually leave. I say usually.

If you want to buy a sandwich or get a coffee you are better off to do it at STP, you will already have noticed the shops to your left as you walked through there. You can do this at King's Cross but most of the outlets are on an upper level with escalator. There are free toilets at STP (which is unusual in a British mainline station) but they are up two irritating small steps which impede wheelie bags.

It takes about 2:20 according to the National Rail website.

Edit 27 February 2018

With the opening of a connection between Thameslink and the East Coast Main Line there is now one direct train per day between Gatwick and Cambridge. It currently leaves at 1203 arriving 1414

The other way you could go is via Liverpool Street station.

1. Take the Thameslink to Farringdon.
2. Cross over to the tube side of the station. There should be lifts. (Lifts for return journey are out of action at the moment but it's only about 5 steps.)
3. Take any eastbound tube to Liverpool Street Station
4. At Liverpool Street Station take a train to Cambridge. It's quite a regular service.

This route would have less walking than via St Pancras/Kings Cross which is quite a more than a quarter of a mile distance.