Do I also need a multimeter if I buy a digital oscilloscope?
I agree with the others, you will need both.  

A digital scope is optimal, but depending on funds available bear in mind that you will get a wider bandwidth for the same money with an analogue scope.  

For example the DSO nano v2 has a 1Msps sample rate which means it will only be able to display any signals reasonably up to around 200kHz. It goes up to 80V p-p.
Beware of adverts for digital scopes mentioning e.g. 20MHz analog bandwidth, check the real time sample rate and divide it by 5 to get a reasonable idea of the highest frequency you will be able to display usefully. If the scope has ETS (equivalent time sampling) you will be able to see higher than the (real time) sample rate for repetitive signals and make use of the analogue bandwidth.
To give an example of a misleading advert (conveniently using the DSO nano adverts) note on this page it says 1MHz analogue bandwidth, but in this page it says 200kHz (1 Msps). You have to wonder whether that's a genuine mistake :-)  

In comparison, for the same price as a DSO nano v2, you can probably pick up a 100MHz bandwidth analogue scope (500 times the bandwidth of the DSO nano) which can be used at up to maybe 400V p-p. I just looked on eBay and picked one at random. People are almost giving away 20Mhz analogue scopes (still 100 times the DSO nano v2 bandwidth)
You will miss out on a few useful features that digital scopes have (storage, pre-trigger capture, etc) but if you are working with microcontrollers you will struggle with 200kHz (e.g. even a simple PIC16F may be running at 16MHz with SPI/UART/I2C faster than 200kHz) 

Either way, a bad scope is better than no scope. Shop around a bit, if you can find a decent DSO within your price range that has the bandwidth to cope with what you expect to be working with then grab that. I would try for something with at least 10MHz bandwidth (so around 50Msps for digital)
Check out the Picoscope range for PC oscilloscopes, they are pretty good from what I hear.