Lime Tree to Canterbury

It would have helped if you told us where this hotel is. Lime Tree hotel is a common name.
Is it in London or somewhere else? I found this website for a hotel of that name in London: http://www.limetreehotel.co.uk/
Address: 135-137 Ebury Street, London, SW1W 9QU

With that information I can look things up. That address is near Victoria station, nearest tube stations Victoria and Sloane Square (about equidistant). So the easiest option is to walk to Victoria station and catch a train.

Again, this is assuming you mean the Lime Tree Hotel in Ebury Street, London. If it is a different one, please say.

Sorry but it is Lime Tree Hotel on Ebury Street in London.

You can also catch the coach (bus) from Victoria Coach Station, which is right across the street from the train station, on the west side. The coach station is closer to the Lime Tree than is the Victoria Train Station. Coach (bus) is cheaper, comfortable, has wifi, and you will see more towns and villages as you pass through on the way to Canterbury. The train is the fastest way, but it's in a "cut" some of the time, so you won't see much scenery.

Note that on this map, the Lime Tree is just around the corner, and half a block from the coach station:
https://www.google.com/maps/place/Lime+Tree+Hotel/@51.4928659,-0.149048,17z/data=!4m7!3m6!1s0x0:0x412d8ed0dffe33a5!5m1!1s2018-06-17!8m2!3d51.4934003!4d-0.1499921

Walk-up same day fare is 13.50 pounds per person. I put in today's date, and that's what I got for a fare.
You can check fares and timetables at www.nationalexpress.com

Journey time: 1 hour 55 minutes.

Several (10-12) buses per day to Canterbury. You could walk in, buy a ticket, and simply take the next bus out.

"Anytime" rail fares for tomorrow are in the £32 - £42 range. Your travel date isn't lpaded yet on the nationalrail.co.uk website, but for Friday, Aug 31, I see an £11.30 Advance fare (probably non-refundable/non-changeable) for a train departing at 10:10 AM. That same fare is available for several later trains on Aug 31 as well. I assume you'd be able to buy tickets at that price for Sept. 14 if you purchased them soon after they went on sale, which I imagine will happen in less than 2 weeks.

You would do well to learn to use the National Rail planner to see your options.

http://www.nationalrail.co.uk

The train from Victoria station to Canterbury takes between one hour and 1.5 hours. Advance tickets (purchased in advance for a specific train) are 11.30 GBP to 13.60 GBP (the latter for the fastest direct train).

Walk up prices are 27.10 for off-peak travel, or 32.60 for “anytime” travel.

Hey!! You are following in my footsteps!! Just did this part of your journey too. We went by train. No problem walking to Victoria Station from the Lime Tree altho it’s a few blocks away.

However, next time I might do Rebecca's suggestion and use the coach as there was very little to NO luggage storage on the train cars we were in. I think it must be geared more toward commuters. We finally realized we could slide the cases under the seats but that was a bit awkward. On the coach the driver stows them underneath for you. The coach station and Canterbury East Rail station are very convenient to the tour hotels.

If you opt for the train watch the signboards at trackside. Our train split along the way so you need to make sure you are in a car that continues to Canterbury. No big deal, just be aware.

We went out on Sunday before the tour started on Monday and appreciated the extra time in Canterbury.

National Express coach quoted times usually assume very little traffic. You may end up taking much longer than expected, especially on routes like those out to Kent where the motorway doesn't start close to central London.