Traditional Muay Thai punches were initially very limited with straight jabs and circular punches being the most utilized strikes. It all changed with a globalization of Muay Thai boxing. Western boxing strikes started being adopted by nak muay farang - foreign practitioners - and introduced new kind of punches and combinations.
In my old gym, I have trained both: traditional Muay Thai punches and Western boxing punching techniques. The more esteem was put on western boxing techniques though. The reason is simple: boxing strikes are more powerful than a traditional Muay Thai strikes. That said old school strikes can be more suitable for setting up a kick or an elbow or knee strike right after the punch. The traditional style also definitely utilizes stiff jabs more often. Western style is more focused on delivering powerful punch combos while Thai style put more esteem on balance between punches and kicks.
Most of Muay Thai taught in western gyms, especially those focused on preparing adepts to compete on a pro/semi-pro level, is a hybrid Muay Thai Boxing. An improved and more effective version of a traditional art. Thanks to this traditional punching arsenal got enriched by hooks, uppercuts, jabs, overhands and shovel hooks.
One more important thing to do before you start training is getting the right gloves. Don't forget to protect your hands and wrists properly while doing a heavy bag work and sparring. I know from the experience that wearing a proper gear makes a lot of difference.
Jab's main purpose is to annoy and distract your opponent and to measure the distance between the fighters during a fight. In attack, you can use it to compromise the other guy's guards and open the way for the main weapon: a cross punch, a high kick or a combination of both. It is also very effective as a defensive weapon against a charging opponent - with the right timing you can stop him half way and unload something punishing in return.
This is you money punch. Use it to attack your opponent's face and torso. Relocate weight from your rear foot to your front foot and thrust your shoulder forward. Twist your hips to maximize the punching power. To throw a cross punch you need to build a set up first. The most utilized one is to start with a jab and then follow with a cross. You can also start with a left low kick - if you're right-handed - and then land a punch with your right hand.
It targets a face, a chin especially. It is a powerful close range weapon. Useful only at a very close range - not used in traditional Muay Thai fight due to neck wrestling and other clinch techniques. It is more popular in K1, cause the rules allows only for a minimal clinching (and no elbows). If telegraphed may be countered with a cross punch.
Use it to target gaps in your opponent's defense. When you see an opening, when his guard is low it's a right time to throw this punch. Aim for a chin cause this is a "natural" target for swing punch but it is also useful to deal the damage to the body, rib cage especially. If the other fighter keeps his guard up, a few blasts to the body may convince him to lower his guard to protect it and to expose his face in the process. It is a mid range attack that possesses an explosive knockout power. Twist your back and shoulders to increase the punching power.