There are several reasons to spend more money on a multimeter.
Safety. If you are doing work involving "mains" electric supplies then you should be aware of "overvoltage categories". If you are only doing portable appliances then "CAT II" should be ok, if you are doing installation work you really want "CAT IV". I would also avoid anything not bearing the brand of a reputable western company as it is all too easy to fraudulantly claim safety compliance.
Features, do you want a current clamp? frequency measurement? an interface to connect to a computer? a crude but sometimes useful osciloscope?
Precision and accuracy, this is where multimeters can get REALLY spendy. The more digits a meter has the more accurate and stable all it's internal components need to get and the more attention needs to be paid to noise and distortion.
Regarding leads/probes, most multimeters use standard 4mm shrouded sockets (some very cheap ones use a non-standard semi-shrouded 4mm, old ones used unshrouded 4mm or even 2mm). If the leads/probes that come with your meter are not appropriate for the work you are doing feel free to swap them out for something else.