Most of the all in 1 apps are not really that good when it comes to the odd DVDs (and there are a lot of those).
Therefore I favour a more complex approach, it takes about 5 minutes setup time all in all but.
DVD Decrypter +ripit4me frontend (search around doom9 forums or if you have to I will sort out a mirror: http://forum.doom9.org/ ) works most of the time and DVDFab takes the few that do not. http://www.free-codecs.com/download/DVDFab_Decrypter.htm
Now some nice vob, ifo and bup files appear. Here I would reburn or use DVDShrink/rejig, voblanker and all the other assorted DVD tweaking tools but we are going to AVI (not to mention that is a whole other area).
You want something to pull apart the vobs into something nicer.
DGMPGDec is it: http://neuron2.net/dgmpgdec/dgmpgdec.html
This will allow you to make an avisynth script (not hard once you see how it is done, the helpfiles there are great for the initial stuff)
You can add a resize and filters here if you want but most DVDs do not really need it (if going for a standalone check what resolutions and other stuff may be needed, XBMC is pretty bulletproof if you are going that route).
Now you can do just about anything with it when it comes to encoding it, I like megui but there are some more simple apps if AVI is all you want.
Megui can be grabbed from http://x264.nl/
Virtualdub in pretty much any of its incarnations will do here too if you can find codecs for it.
QuEnc is also good (I use it mainly for DVDs and DPG but it does other stuff too): http://www.bitburners.com/quenc/
And if you happen to reburn to DVD this tool is great for tweaking subtitles (I use it mainly to get rid of the insipid yellow used on a lot of anime DVDs and typos (or use of US English) but it does get far more in depth if you need it to): http://www.free-codecs.com/download/DVDSubEdit.htm
Edit: To add to CockroachMan gordianknot is a bit old but it is rock solid (its more automated cousin autoGK actually mirrors my own methods above fairly closely).
DVDs are based on a specification and a lot of them (very often including stuff from the majors) pays little attention to this. Anime DVDs (although R4 are normally fantastic: to the end that I import them on occasion) are especially bad.
This then makes some of the simpler tools encode badly at best or fall over at worst when trying to convert.
Odd things can range from mastering/authoring/muxing errors (more examples than I care to list: ever had your software player error out when you click various menu options)
to odd pulldowns (simply put DVDs are broken into half frames called fields (ever seen horizontal black lines in videos: this is what happens when you do not convert properly, see deinterlacing for more on this) and these played multiple times to make it up to NTSC standards but some companies use the oddest methods of doing this making for at best jerky playback
odd aspect ratios (BBC DVDs are notorious for this and calling it widescreen). When simple tools come to resize their maths gets messed up and you end up with a bad video (odd size: hard to encode: lots of space needed and if there is none available quality takes a hit).
audio encoding issues: there is the old PAL MP2 thing (it is not standard for NTSC but nobody since the very first DVDs really uses it) but there are little things and odd bitrates can trip up some apps. Bad sync when ripping can also be a problem: sync can be fairly easily tweaked on DVD but ripping apps sometimes forget this.
video quality: not so much a problem your end except when it comes to encoding (errors and by definition unusual which means space is needed to store them)
Just to add another point for do it yourself flexibility: I might want to make a 1.9 gig AVI file but a lot of the fire and forget types are 700MB (or some iteration based on CD size) or get lost (credit where credit is due though it has got far better these last couple of years). They are frequently also set bitrate rather than quantisation (ever wondered how some scene rips look so good and are odd sized: now you know) As I dare say most people use DVDs these days 700 megs is somewhat pointless (if you can ignore the previous line to an extent (every film is unique and needs treatment to that end) I find 700MB and the associated bitrate a shade too small for most films but 2x700MB is a bit like overkill for most films. A gig and bit (which coincidently means 4 films if you do it right to 1 single layer DVD) is normally spot on.