Ambleside, its far easier for public transport round the Lakes. Bowness is also unpleasantly busy, its just a traffic jam.
Keswick is an option, it depends on your interests as to whether its a better option, Ambleside is better for traditional and obvious tourist attractions, Keswick is better for quiet and relaxation. Both have merits if you want to do walking.
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If you are travelling by train, Kendal makes a great base.
The train goes to Oxenholme the Lake District (on the outskirts of Kendal).
There is a branch line that runs from Oxenholme, through Kendal and on to Windermere. Kendal station is central to town so you would have all the restaurants, etc. of the town in easy walking distance. You would also have easy access to other parts of the county from the bus and train stations in Kendal.
Plus it's a lovely town with lots of character. :)
Ambleside for sure (less busy :) )
Ambleside.Its a lovely place with cafe"s,restaurants,pubs,shops and good walking area.
Personally,i would choose Keswick as my base if i was a first time visitor to Lake District.
Ambleside is still primarily a base for fell walking, as others have said it is much quieter than Bowness, but it does have some good restaurants and cafes and a couple of decent pubs.
Bowness is more of a holiday home and day tripper place, and as people have said it does get absolutely packed on summer weekends and bank holidays. A consequence of this is that some of the pubs (especially one with tables outside) have started charging ridiculous prices for drinks and pub food. one is charging £18 to £22 for a steak, the Bistro literally across the road from it (and which is very good) does a set meal that includes steak for £16, crazy.