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.
Yes!
Buy both.
They are complementary in functionality and you will never regret having both.
A multimeter is your basic tool. Utterly indispensable.
Having several cheapish multimeters for everyday work and knowing their limitations is a good idea. Know about how accurate they are for each sort of measurement. Have an approximate idea of the input resistance. Know the resistance of the current ranges (most people don't, it varies quite widely and it can matter). [[I probably own more than 20 multimeters :-). Most are cheap ones which allow multiple simultaneous metering of experimental setups.]]
A high accuracy or extended digit count multimeter is a luxury. You want it if you can afford it but you can do without it.
An oscilloscope is your heavy artillery. It can do things that a meter can never do. It gives your brain the ability to visualize things happening in the time dimension. It is an utterly indispensable tool for anyone even vaguely serious about electronics. Even quite a bad scope is better than no scope - but a half good scope is far far better.
Scope add ons for PCs are great. They provide value for money performance not readily achieved by other means. BUT it is very very hard to beat the physical space division twist a knob and push a button interface of a more traditional oscilloscope interface. Even modern scopes which are all electronic use a mechanical interface with substantial resemblance to a traditional one.
Multimeters are far more accurate than oscilloscopes. This is what you gain in exchange for living with near-DC only. Oscilloscope channels usually have only 8 to 12 bits of resolution, which is like having a 2.5 to 3.5 digit meter. Also, scopes generally can't handle high voltages; you have to get special (read: expensive) probes.
Also, multimeters can measure things like resistance, current, capacitance, temperature, and diode voltage drops easier than a scope. Some multimeters also have neat features like min/max and true RMS AC.
Truth is, you will probably need both.