I would stay clear of the starter edition honestly. It makes unlocking an operator way more of a hassle (it costs way more to unlock one in the starter edition than the standard version), making progression more tedious than it is normally.
Play through all the situations (short "missions" that showcase some of the operators, which also nets you a good chunk of renown to use for unlocking operators and attachments for your weapons) and play some terrorist hunt as a way to get familiar with mechanics before heading into casual. Once you unlock Ranked play at level 20, leave casual behind.
You sadly can't try out operators before unlocking them, so here is a short list of the operators you should unlock first (this list is my opinion! some will disagree, and I welcome changes!)
Defenders:
- Rook: set down a bag of armor plates, and your job is done. He also has more armor than most other operators, so he is a bit more forgiving for newer players. Headshots are still headshots though.
- Jaeger: Gives you three anti-grenade "trophy systems" for you to place around the objectives, or wherever you deem it appropriate. He also has solid weapons that are easy to use. Fast as hell, and works just as well in and around the objective as he does on his own roaming the rest of the map.
- Bandit: Has three batteries that you can attach to reinforced walls to electrocute them (this prevents anyone being up-close to said wall, or any gadget being placed), on barbed wire or on placed shields. Plays fairly similarly to Jaeger. Note that his batteries will destroy friendly gadgets if they are close enough.
- Frost: (DLC Character) Has three bear traps you can set down causing people to enter a downed state while being unable to move. Works great as both a roamer and a objective defender.
Attackers:
- Sledge: Carries a big fucking sledge you can use to create doors in walls that haven't been reinforced. Can also be used to knock down Castle's barriers without the use of grenades or brute force. Has pretty good weapons, and won't ruin a round if he dies early.
- Twitch: Can throw out a shockdrone that can hurt other players as well as destroy gadgets (such as badit's batteries etc). Has a very good weapon in the FAMAS, and is very fast.
- IQ: Has a display on her forearm that can spot any electronics within 20(?) meters. Great for learning camera locations, spotting sneaky c4s and for locating Pulse, among other things. Moves fast and has great primaries.
- Blackbeard: (DLC Character) Has a freaking shield in front of his primary weapon, making him harder to take down than most. Great for locking down sightlines. Just remember that his shield isn't very large, so moving just a bit can open you up to some nasty headshots when you least expect it.
Some people will probably suggest either Thermite or Fuze as valid operators to unlock early on, but I feel like they can do more harm than good when in the hands of someone who just started playing.
I tried to focus on operators that help you notice the layout of the map and familiarize yourself with where to put what, rather than being the backbone of a make or break strategy.
As for the DLC, I'd say hold off until you know if you actually want to keep playing. It is absolutely worth it, but you can still unlock everything the DLC includes without purchasing it. It just takes time to accumulate enough renown to be able to buy the new operators (maps are free for everyone).
This went on for a bit longer than I had intended, but hopefully it is useful to somebody