You have my support for starting with ATWQ. Most likely you want to have a new experience with the ASOUE universe and starting with ATWQ is a good way to do this. But in addition to that, I also suggest other things that may seem strange, but I find cool ways to read this books:
(It's not exactly chronological, but it should come close to that ...)
1 - Read LSTUA page 196 and 197, TBL letters LS to BB # 1, # 2 and LSTUA page, 52, 168, 169 (first text) 189,190 and 191.
2 - Read ATWQ ?1,?2,?3.
3 - Read LSTUA page 61-65, 165 and File Under 13.
4 - Read ?4
5 - Read that lemony tale with the emperor baby I forgot the name but I ask someone here to help me and leave the link.
6 - Read LS to BB # 4
7 - Read LS to BB # 3 and Read the poem My Silence Knot.
8 - Read Lemony's letter to Dr. Charley Patton (found in LSTUA chapter 1)
9 - Read the newspaper articles in chapter 5 of LSTUA and the letter from the vineyard to Lemony in the same chapter.
10 - Read chapter 6 of LSTUA.
11 - Read LS to BB # 5
12 - read LSTUA pages 33 to 51.
13 - read LS to BB #6
14 - Read The Dismal Dinner.
15 - read LSTUA Pages 176 to 178, 182 to 183,
16 - Read LSTUA Chapter 4.
17 - Read LSTUA page 150 and 151,TBL BB to LS#2, then Read TBB the rare edition along with the notes (You do not need to read normal TBB) and TRR.
18 - Read LSTUA page 162, 134 to 137, and TWW.
19 - Read , LSTUA Chapter 9, LSTUA chapter 1: the newspaper article and the personal note.
20 - LSTUA Pages 117 to 120.
21 - Read TMM.
22 - Read LSTUA 127 to 131 and 142.
23 - Read TAA, TEE and TVV.
24 - Read LSTUA Pages 132 and 138 to 141.
25 - Read THH, TCC and read the rest of LSTUA starting with the introduction.
26 - Read TBL BB to LS#3 then Read TSS.
27 - Read TGG and TPP.
28 - Read TBL BB to LS #1, #5 and "To My kind Editor.
29 - Read booklet 13 Shocking Secrets.
30 - Read The End.
31 - Read Poison For Breakfast.