I will provide the counter argument to the above.
I would suggest considering a mid range older bike that doesn't need a lot of structural replacement (so as to keep costs down), if the goal is to have fun restoring and tinkering.
Many people do restore high quality bikes but sometimes they are "dream" bikes a person could not afford earlier in life or a favorite that racers they followed from the past may have ridden. If you don't have that connection to those bikes the additional cost and limitation of design options may not be worth it for just riding around. The main problem with that is restoring a quality bike costs a lot more money and the range of parts is limited to what is expected for the make. You wind up sourcing specific parts for specific groupsets. Of course that may be fun in itself and you can do what ever you want but you will get a lot of flack if you start stripping certain paint or decal schemes and/or using different quality drive train parts willy nilly.
As far as tinkering and enjoying the act of rebuilding I find mid range older bikes to be fun. They can be had for dirt cheap (or free), they usually were pretty good quality in their day and replacement parts are cheap if needed. The systems are also basic enough that you won't feel nervous messing with them. Because they are not "grail" bikes people will also be a lot more forgiving of any modifications/artistic license you may take in restoration. This is especially true if you go for the extra challenge of finding a real junker to restore. I guess the difference I might say is whether your goal is to have fun tinkering with a bike or whether the goal is to wind up with a quality bike. Though a quality bike can also be fun to tinker with (especially if you have a personal connection to it) the price to play will be a lot more.
Another option is to choose a bike of some historical note or from a well known company. If you just like riding around greenways a more upright utility bike may be what you desire and a great company for those was Raleigh. They have a long long history you can spend time learning about and a used older model can be had fairly cheaply. They are also a very pleasing eye catcher when done up nice.
I have a department store bike from the Hudson's Bay Company called a Hurricane Deluxe 6 which is nothing special as far as quality goes but has been a real pleasure to restore with inexpensive period correct parts and ride around. I also have a low quality 10 speed that I felt free to strip and rearrange into a single speed at the cost of some tires, a $15 sprocket, chain and a lot of elbow grease. Another bike is a mid 80's mtb junk store find that I explored using a different finishing technique on (Gun Blue treatment instead of paint). Lots of fun and creativity at little relative risk or cost.
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