Second hand 'pure' electric cars in that price bracket means Nissan Leaf and Renault Zoe. Teslas are available but are much more. There are a few oddball cars around (such as the Mitsubishi iMEV and its Citroen and Peugeot clones) and some that sold in very low numbers (Renault Fluence and previous e-Golf), but I would tend to avoid them because of the difficulties in any low volume car in getting parts.
Be careful with Zoes. Initially, Renault sold the car but only leased its battery. That didn't make too much difference new, when a lot of people bought them on PCP, but secondhand the lease is quite a big monthly payment. Later cars had the option of buying the car outright. Battery leasing came with a promise that that battery would be replaced as and when it's capacity dropped below a certain level, but the impression I get is that batteries are lasting better than expected and so very few are being replaced under guarantee. There were some Nissan Leafs sold on this basis too they are called 'Flex' and dealers don't always make that clear in their adverts.
The Nissan Leaf had a significant mechanical/electrical upgrade in 2013 when they moved production from Japan to Sunderland and generally the Sunderland-built cars are considered better; the main difference is that the Japanese cars had a light interior but nearly all the UK cars have a black one (a very few had the light one as an option). Also avoid the lowest trim, called Visia or Visia+, because the general view on those is that the cost cutting went too far -- for example, in the 2013 upgrade they put a much more efficient heater in the other trims, but left the Visia with the original one.
All electric cars are automatics (strictly, because the electric motor is more flexible than a conventional one, they only need a single gear so there's nothing to change -- even reverse is by running the motor backwards).
The main snag is range. Newer cars are getting better. Leafs and Zoes should comfortably manage what you describe, but long journeys require careful planning and relatively frequent recharging.
The other snag is the effective battery life. The problem here is that the cars aren't old enough for anyone to really know how long a battery really lasts. There are some horror stories from the US, but temperatures can get much hotter and colder there than here, and batteries don't like extreme hot and cold. In the UK batteries seem to last reasonably well.
Real charging costs depend on where you charge. The standard unit of electricity is the kiloWatt hour (kWh), which is what your electricity meter measures in. The equivalent to mpg is mpkWh. A Zoe or Leaf will probably achieve between 3 and 5 mpKWh, depending on how it is driven and the weather -- performance is worse in rain and cold (which is also true for conventional cars but gets hidden because their fuel reserves in terms of miles are so much greater). At home I am currently paying 12p per kWh. In contrast, DW's 1.6 diesel estate does 45mpg, which is 10 miles per litre, and diesel round here is around 120p per litre, so I am getting 4 miles for the price of 1.
If you charge away from home, then prices are more variable. Greater Manchester, where I go quite regularly, has lots of free public charging totally free (though you pay £20 a year for the access card there is a free phone app but that's too flaky to rely on). At the other end of the scale, Ecotricity at a motorway service station will charge a £3 access fee and then 17p per kWh, which can work out being more expensive than diesel would be.
Do you run out and get stranded? Not yet. The cars are very good at monitoring how much charge they have, and there are several apps that will find the nearest charge point. The longest trip I have done in the Leaf, which was in summer and driving carefully, was about 84 miles return. If you do, then the AA or RAC should recover you to a charge point.
Servicing is different. Most of the servicing on a conventional car oil, air and oil filters, exhausts, spark plugs isn't needed on an electric car. There are some people out there who think you don't service an electric car once its out of guarantee because there's no point. Even brake wear is minimal because most braking is electromechanical. They do of course have other things that need monitoring and changing -- you still have tyres and brake fluid and wiper blades for example. Equally, there's a lot less to break, but you still have suspension and body and quite a lot of electronics. My last breakdown in a conventional car was when one of the front springs broke, which could just as well happen to an electric car.
Electric cars have tended to be placed in higher insurance groups than equivalent conventional cars to date, but otherwise it's very difficult to say what the insurance costs are. I am an 'experienced driver' with 9 years no claims living in NW England and I have just renewed on my Leaf for 15,000 miles a year at £270.
The best way to decide is to get behind the wheel and test drive one. When you do, you'll find that nothing this side of a Rolls is as smooth and as quiet.