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While you don't need to constantly look over your shoulder or be suspicious of everyone you see, as a tourist you should take sensible precautions:
If you are eating at a cafe or somewhere outdoors, don't leave your wallet or any other valuables on the table. Keep items like your camera safe when you are not using them. An opportunist thief or pick pocket could take them without you noticing.
In Salisbury, being targeted by petty criminals isn't really a problem: it's essentially a small town and a welcoming city for the tourists it gets: probably attracted by Stonehenge, they will find mediaeval buildings, and beautiful countryside to make a longer stay worthwhile. And then for the tourist it's a relaxed atmosphere: prices are fixed, there is no haggling, there are no ticket touts or others to harass the innocent visitor.
The rail station is convenient, and Salisbury is small enough to explore on foot. If you are visiting by car, then take care of parking restrictions. The traffic wardens, uniquely called 'Ambassadors', will catch you. When they are not issuing parking fines, Ambassadors (easily recognised in their black & yellow uniform) are a useful source of local knowledge and will help you find your way around.
Places to avoid at night are Fisherton Street and Milford Street, where there are a few nightclubs, although all the city centre streets are monitored by a CCTV system. Also avoid the grounds of the Arts Centre late at night.
If you ever do get in trouble you can go to the local Police. And if it is very serious you can of course go to your country's Embassy (in London).