I will just include the mandatory recommendation here to get a patent attorney. If your invention is worth protecting, it's worth protecting properly. Otherwise, you're just throwing money away at filing fees with little chance of getting something at the end.
With that out of the way...
US Non-Provisional
What must a US non-provisional application include?
In short, you must have a description, a claim, and a drawing at minimum.
More formally, the requirements of an application are provided by 35 USC 111(a)(2):
(A) a specification as prescribed by section 112;
(B) a drawing as prescribed by section 113; and
(C) an oath or declaration as prescribed by section 115.
The specification comprises (35 USC 112(a)–(b)):
(a) IN GENERAL.—The specification shall contain a written description of the invention…
(b) CONCLUSION.—The specification shall conclude with one or more claims…
Can claims be added later?
Yes.
MPEP § 506(I)(C) provides:
Filing dates are accorded to nonprovisional applications filed under 37 CFR 1.53(b) submitted without … at least one claim (for applications filed on or after December 13, 2013) …. In such cases, a notice is mailed by OPAP requiring the appropriate fees, claims (for applications filed on or after December 13, 2013) and for applications filed before September 16, 2012, the oath or declaration (which must include the names of all the inventors), accompanied by a surcharge (37 CFR 1.16(f)).
You will have to provide the claims shortly after though.
If I sell publicly after filing only specifications, but before filing my claims amendments, will that affect what I can claim?
No. If you sell the application such that the applicant is assigned to another party, then the obligation to add claims passes to whoever the new applicant is.
PCT application
What must a PCT application include?
PCT art 3(2) provides:
An international application shall contain, as specified in this Treaty and the Regulations, a request, a description, one or more claims, one or more drawings (where required), and an abstract.
Can claims be added later?
Not really.
PCT art 11(1) provides:
The receiving Office shall accord as the international filing date the date of receipt of the international application, provided that that Office has found that, at the time of receipt:
…
(iii)…(e) a part which on the face of it appears to be a claim or claims.
The consequences are provided by PCT art 11(2):
(a) If the receiving Office finds that the international application did not, at the time of receipt, fulfill the requirements listed in paragraph (1), it shall, as provided in the Regulations, invite the applicant to file the required correction.
(b) If the applicant complies with the invitation, as provided in the Regulations, the receiving Office shall accord as the international filing date the date of receipt of the required correction.
Thus your PCT application will be dated as of whenever you supply claims. If this is later than 12 months from the earliest priority claim (your US provisional), you will be too late: your priority claim will be invalid.
Commentary
Thus there is some scope for not including claims in the US non-provisional immediately. However, this is a really bad idea.
When drafting your application, you should almost certainly start with your claims, and craft your description around the claims. This ensures that the description provides the necessary support for your claims.
Doing it in the other direction is risky. You would have to hope that the description contains the terminology and the features in a suitable manner that they can be used to form supported claims. While the US is somewhat forgiving on this, other jurisdictions (notably Europe) are much more restrictive. I have seen far too many applicants make this mistake, and get left with good inventions that will never get a patent due to poor drafting.
