I wasn't there (I was born in '67) but I've read quite a bit about the war, especially on the small unit level.
My understanding is that pretty much *anything* made up 'till then turned up in Vietnam at one point or another.
The early U.S. advisors used a lot of U.S. WWII weapons, as did the ARVN troops the U.S. was advising. A lot of M-1 Garands were rebuilt in U.S. arsenals to keep the supply of foreign military aid to Vietnam flowing. The Garand was particularly ill-suited for the Vietnamese because many were of very small stature.
Later when the Garands were replaced with AR-15/M-16's, the Garands still in the rebuild pipeline were retained stateside. Many were rebuilt, put into storage, and never issued again. Some were later released under the DCM or CMP programs. I have one that went through that 60's era rebuild that I got from the CMP.
Early AR-15's were issued to U.S. advisors and Vietnemese troops before the U.S. military even planned to adopt the weapon itself. The AR-15 was smaller and lighter and had less recoil then the Garand, so it worked better for the Vietnamese troops. The suitability of the AR-15/M-16 as foreign aid to Vietnam was one of the things that kept that rifle program alive early on. The U.S. military didn't want it for itself, but was willing to let the ARVN troops use it.
The later models, the XM-16E1, USAF M-16, and US Army M-16A1 became the standard U.S. and ARVN weapon in the mid-sixties, largely replacing the M-14 (in U.S. service) and the M-1 Garand and M-1 Carbine (in ARVN service). The USMC hung onto their M-14's past the point were the army switched over. (I don't know if that was USMC preference or because of a lack of M-16's to arm the USCM. I suspect a bit of both).
The early U.S. advisors and Special Forces also used a variety of non-standard weapons as well, mainly left overs from WWII. From what I've read, it seems to me that if a U.S. advisor or SF unit got it's hands on something unique they wanted to use, and could get a supply of ammo for it, they'd use it until the ammo ran out or they were ordered not to use it anymore. In addition to the fairly common "Swedish K" SMG (and the U.S. S&W 76 copy), I've seen references to U.S. advisors or SF teams using German MP-40's, Chinese PPsH SMG copies, and other odd stuff.
The VC used anything they could get their hands on, especially early on before they were really being supplied from the North. As the war progressed, their weaponry was slowly upgraded with weapons brought from NV into SV via the Ho Chi Minh trail.
Early Viet Minh and early Viet Cong used mainly WWII era weapons: There were a lot of left over Japanese WWII weapons. They also used a lot of captured French weapons, especially when they were still fighting the French, but even later into the war. We're talking about French MAS rifles, MAT-49 SMG's, and things like that. Many of the captured MAT-49 SMG's were later rechambered in 7.62 (Pistol) instead of 9mm, for ease of ammo resupply.
The Mosin Nagant was another fairly common rifle for the VC early on. I believe these were Soviet aid exports. Of course, the VC being guerillas, would always use whatever they could capture from ARVN or U.S. troops or otherwise scrounge up. So, if the U.S. or ARVN used it, expect for the VC to have some as well.
Later, when the VC started getting regular supplies from the North, they used more Chinese built SKS rifles and Chinese built AK-47's (Chinese Type 56).
When the U.S. started sending in "regular" troops, in large units, instead of just advisors and SF teams, the weapons became more standardized. The early units used the M-14, with the XM-16E1 and then the M-16A1 becoming standard for many units sometime around '65 - '66. It did take time to make the transition, so both rifles were in use at the same time.
The grenade lauchers included the M-79 single shot launcher, an experimental 3 shot launcher that was tested in country (very few), and later an experimental underbarrel launcher that was withdrawn after combat use proved that it was not durable enough. The M-203 underbarrel launcher came around late in the war, after the first failed underbarrel launcher.
The standard U.S. Army pistol was the 1911A1. Army helicopter pilots were often issued a .38 Special S&W, although many acquired 1911A1's later. Army tunnel rates liked the .38 Special revolvers when they could get them, but most probably used the 1911A1 .45 as it was more readily available. The 9mm Browning High Power was a favorite of SF types.
Don't forget the South Koreans, the Australians, and other countries sent troops to Vietnam as well. There was a lot of different stuff used in the war.