Cheapest Set To Complete?

Lincoln Memorial cents, State Quarters (minus the silver proofs, at least), maybe Jefferson nickels (pre-2003). Roosevelt dimes would be nice too, but the early silvers are quite expensive, the 1996-W is an obscure key, and even if you get all the other dates you still won't have a 2017.

If you want something that isn't pretty much current money (though then you shouldn't have posted it in this subforum), the traditional answer is Franklin halves, but that's still 35 large silver coins, which will cost an awful lot of money.
There are very many key dates in Buffalo nickels, but it should be a reasonably cheap set if you're willing to include acid-dated coins (though the 1913 Type 1 would be tricky - people tend to identify it by the type difference rather than the date).

I'd recommend Braided Hair large cents, personally, but they're a bit expensive, especially for the later dates. Though if you look for especially low grades, they might even end up cheaper than Franklin halves, because there's only 19 instead of 35.
If you can find a cheap 1857, in particular, the other dates should be easy.

Some people don't recognize 1856 Flying Eagle cents (claiming that they are actually patterns). If so, the rest of the Flying Eagle cent series would be fairly cheap. The 1856 is very expensive though.
Most do recognize 1872 (and 1871 and 1870) 2 cent coins, so even if you don't recognize the 1873, that set won't be cheap at all. Sorry.

I'd say With Rays Shield nickels (1866-67) are the non-modern set with the cheapest total, ultimately.

Ho, way a minute here. Why wouldn't we be able to get a '17 Roosevelt? Have I missed something in the mint news?

If you're talking easiest and cheapest to complete "from circulation", yeah, I think the Jefferson nickels would be a good one for ya. I know I enjoyed doing mine, and you can conceivably do it from just looking through rolls of nickels (C.R.H. -coin roll hunting)
(But then you want to always be on the lookout for better conditioned ones to upgrade with in there...making a nicer looking set ! -- it never really ends...lol)
Go down to a hobby store like Hobby Lobby or your local coin shop and buy the " Jefferson nickel" coin folder.

I would save F.S. examples for an album with proofs and not a folder though, especially since you just want coins "where you can just make out the date"...

It is not the absolute CHEAPEST set in the world, but 1940-1958 wheat cents. It is #2 to the entire collection.

How much do you "need" all proofs, silver proofs and rare varieties?

Excluding the varieties, a 1959-2008 LMC set can easily be put together for face... total cost of $1.08 if I recall. It took me six months of change-checking to get 1959-1978 in XF and 1979-2008 in AU/BU.

Omitting the varieties, you can get a set of Jeffersons 1938-2003 for like $40 on ebay in VF+, or like $60-100 in AU/BU.

You can easily assemble a full set of clad dimes and quarters from circulation. The 1996-W as stated does not circulate, but is very available online.

Half dollars can easily be assembled from bank boxes if you can get them. You can get the silvers, 70-D, 1987 P/D, and post-2001 in BU for a very reasonable cost if you buy raw and buy the NIFC as a lot.

If you just want to get it all, you can buy mint and proof sets in lots on ebay. People are de-stashing their 70s/80s/90s proof sets constantly, usually for much cheaper than individual sets, and much, much cheaper than putting the sets together one coin at a time.

As you may already have noticed, Jefferson nickels are one of the most easily accumulated sets. And lots from just change. One of the things I like about a Whitman Classic Album is you can put almost all of the Jefferson nickels in one binder. Same with Lincoln Cents. However, a complete Lincoln Cent collection is rather an expensive collection to complete.
Same with Roosevelt dimes as with Jefferson nickels. And again in a Whitman Album, all will fit from the first year to present. Mostly what is nice about those is so many are available in change. I've known many collectors that completed a Jefferson set without buying any of them.

I think everyone has overlooked the most obvious "cheapest" set to complete.

Lincoln Sheild Cents! From circulation this would cost you 14 cents for all of the P and D issues, if you somehow found all of the proof issues from circulation then this would be another 7 cents. Now that's cheap!

I completed a slabbed Susan B Anthony set of PF69Dcam, & MS65 business strikes with all of the major varieties (wide rim, type 1 & 2s) for something like $300. You may want to look into it.....prices are ripe for cherry picking the best examples at the moment! There are only 4 years of the coins, so it's a short set to complete.