For car work on a budget we'll tend to recommend something from the Clarke range. A Clarke 130 will tackle everything on a Mini and will work properly (unlike most new and cheap eBay offerings).
It's worth getting a refillable cylinder of shielding gas in Argon/CO2 mix. Adds a fair bit to the cost, but makes everything much easier and welds much nicer. The gas suppliers list has some options cheaper than the industrial suppliers.
There are a few Cebora pocket mig welders on ebay if you are on a really tight budget. This is a robust machine so you should get a good deal. Also plenty of choices for Clarke machines.
You need something that goes down to 25 / 30 amps. Gas could be CO2 as the cheapest option but Argonshield would be better if you want the rental option and additional cost.
The tutorials and guides on this forum are really excellent.