You can use
this list of some important activities as a reference, but the main PvE stuff for me is mainly:
The Gunrunners Mutant Cave QuestThis is repeatable, and you get a nice cache of big guns to sell (or use) each time. There is enough fun in this quest to justify leveling a few alts to do it more often (the timer is 6-8 hours). Plus you get 6 bonus items at the end, and semi-safe access to an advanced workbench for crafting good gear.
Barter Runs!Gradually get rich by working sales all over the map. Semi-safe in guarded towns, but a thrilling chance for PvP in unguarded towns. Get those caps! Or lose all your inventory.
Exploring Random CavesCan be boring, but can be deadly too, and you find rare items. (use an item highlighter). Fight 3 rats, or 7 large deathclaws, or a band of mercenaries. Bring Psycho and beer. Find alien blasters, bk robes, solar scorchers, enclave armor, etc.
Locker HuntingCaps, gear, blueprints, cute items, even some resources. Broken legs from bear traps, sometimes funny NPC behavior, and sometimes PKs drop in to murder you. Maybe find a vertibird!
Mining and CraftingClicking on rocks for an hour can be boring, so a lot of guys spend 40k for a private mine and put slaves there. However, mining HQ can be fun for PvP if you have a proper HtH approach. And the resources are needed for crafting good gear (and xp). Use the bonus items you craft, and sell the standards on your barter runs. Save up caps to buy expensive implants. Buy cars, blueprints, and whatever items you don't have yet or don't want to craft (like drugs).
CompanionsFirst it takes a while to find a perfect NPC companion (max Endurance and Agility), and then it takes a while to level them up, but you'll spend enough time on the map to find some Special Encounters. Bring them into some PvE fights and they die a lot but sometimes they really help.
Protect Your BaseStart a new alt, with Good Natured. Use him to get your own private base. Get his Speech to 130 or so (easy) and train up 3 NPC companions. Station them at your private base as guards. Then bring a friend there to test your security. Add your friend's character name to your terminal and set him as Enemy so your guards will attack. Find out how well they defend, and how well your friend does against them.
Ares Rocket SiloOkay now here is a very tough place, but it's the most fun imo. It's surface level is perfect for early sniper training, and inside...well it's tougher there.
Experimental BuildsThis is one thing this game is famous for, and will suck the time out of your life. There are many build ideas and combinations of perks, traits, skills, and methods. Start now and we'll see you in 3 years to hear your report.
MoreGlow: chance for rare items including implants, with high probability for some PvP
Farming NPC groups like Enclave, Bounty Hunters, Unity Patrols, NCR Army, etc
Warehouse: chance for rare items including implants, with high probability for some PvP
New Reno Sewers: chance for rare items including implants, with high probability for some PvP