You can play an RPG like D&D without miniatures - but the latter editions pretty much require them for fights.
But if you're cool with just describing things you can do without the figurines. We use figures and a dry erase mat with squares on it, but often when things are simple to imagine we just describe what's going on and stop moving the figures around. If it gets complicated the referee draws the area on the mat and everyone puts their figures where their characters are.
The extreme of using figures too much comes when you say your character is going to sneak along the wall and peek around the corner, and you pick up your figure and walk it along and lean it over at the corner. You really don't need figures in that case.
Also the need for figures increases with the number of objects in the fight. A marker or spell effect that you lay down on the board counts as an object just like a player-character or a monster. Imagine a fight between a single PC and a monster. It would be ridiculous to use figures. Then imagine two PCs and two monsters. You can still get away with no figures. More than that and it gets difficult to juggle everyone's relative position. I'd use the "seven plus or minus two" rule here but lean to the low end of the scale.
That means any RPG can be played with just pencil and paper if you have just one or two players. If you have three or four players you should get some tokens to represent them - perhaps bits from other board games for the PCs and dice for the monsters so the referee can track what happens to "monster number 6" for example.
Beyond four players it gets difficult to remember what token represents which player. At 5+ players, and perhaps as few as four, you really need to start using figures.
As for wargame suggestions, you should read up on various rule sets. I'd suggest getting a grasp of the basics of Tunnels and Trolls, Dungeons and Dragons, GURPS, and Shadowrun (or any of the World of Darkness games that use dice pools), Dogs in the Vineyard, and Burning Wheel. That should give you a good foundation of some common types of task resolution.
In terms of a concrete thing you can take away from this post, how about this:
1: Turns
Everyone has to take turns. Flip a coin to see who gets to go first. If there are multiple players, each should do (number of players +1) coin flips and the person who got the most heads goes first, then the one who got the next highest number of heads, etc.
When it's your turn you can pick a unit, have it move and shoot and move if you like, then do your next unit. You can use all your units in your turn if you want.
2: Movement
A unit gets to move once per turn, up to its full movement. If you want to make the unit change its facing you use up some movement to do it. Otherwise it has to go straight ahead. When you turn, you put your finger down on the middle of the unit and turn it.
3: Attacking
A unit gets one attack in a turn. When it attacks you put the bottom of your fist on the unit and flip a coin in the direction you want to shoot. Where the coins lands is where you hit.
Anything the coin rests on is hit. It's possible, though unlikely, for a coin to land on two units at once. For tank shells and plane bombs you hit everything on the ground within 3 inches.
If a unit is hit, the unit's player flips a coin. On tails, the unit is destroyed. On heads the unit survives but it loses its next turn. If a unit dies crumple it up but leave it there - other units can collide with it.
4: Units
Every unit is drawn on graph paper and cut out. Units can be tanks, trucks, planes, or blocks of infantry.
Tank
Move: 5 inches
Turning Cost: 1 inch
Ammo: 20 shells / 10 bullets
Special: The shell hits everything on the ground within 3 inches of the coin. Tanks are immune to bullets. They also flip two coins when being attacked and are destroyed only if both come up tails.
Truck
Move: 8 inches
Turning Cost: 1/2 inch
Ammo: None
Special: An infantry unit standing on it moves with it for free. Or they can load a Cargo onto it. A truck can't carry both.
Infantry
Move: 3 inches
Turning Cost: 1/2 inch
Ammo: 10 bullets
Special: Can ride in Trucks. If adjacent to a Cargo they can pick it up and carry it. They can carry only one Cargo.
Plane
Move: 12 inches (3 inches on the ground)
Turning Cost: 2 inches
Ammo: 6 bombs / 10 bullets
Special: The bomb hits everything on the ground within 3 inches of the coin. Instead of flipping, drop the coin from directly over the plane, from the height of your knee.
5: Collisions
If a plane collides with anything, the plane is destroyed and the other thing takes a hit.
If a tank or truck collides with infantry, the infantry takes a hit.
If a tank collides with a truck, the truck player flips a coin: on tails the truck is destroyed.
If Infantry collides with another Infantry, the infantry that moved hits the other one without using up ammo.
When you collide with something you lose the rest of your movement that turn.
6: Cargoes
A Cargo can be an objective (Blue Team needs to somehow get the Cargo to a certain building, or the edge of the map, for example). But other Cargoes have special effects. Infantry can pick up or drop a Cargo instead of attacking. They can drop the Cargo anywhere adjacent to them, including on a nearby Truck.
Ammo Box: Any unit adjacent to an Ammo Box can loot it instead of attacking. It regains its full Ammo. The Ammo Box is removed.
Medical Supplies: A crumpled infantry unit adjacent to a Medical Supply can use it to get back into fighting shape instead of attacking. Uncrumple the unit or just replace it with a fresh copy. It retains its old ammo level. The Medical Supplies are removed.
Anti-Tank Rocket: If carried by an infantry unit, that unit gains one shot of Tank Shell. When the shot is fired, remove the Anti-Tank Rocket Cargo.
Anti-Air Rocket: This works just like an Anti-Tank Rocket, except it only affects airborne units.
Dynamite: An infantry unit can arm the dynamite instead of attacking. The dynamite blows up in a 3-inch radius. Put a stack of 1-3 coins on the dynamite when it's armed (the arming player chooses). The next time the arming player takes his turn, take off a coin before he goes. When the last coin is taken off the dynamite explodes.
Firefighting Gear: Infantry carrying this gear in a building add one coin to the flip for whether the building becomes Ruined. All flipped coins must be tails for the building to be Ruined.
Tank Trap: Infantry carrying this gear can spend their attack to put it down in a street. Infantry can move through a Tank Trap but other vehicles collide with it.
Debris:
7: Buildings
Different buildings can have effects on the game rules. Unless otherwise noted, Infantry can enter every building but they are the only ones who can enter any building.
Anything in a building is safe from attack until the building is Ruined. A building can be Ruined with any explosive. The attacker flips a coin: on tails the building is Ruined. Otherwise it's fine. A Ruined building gains 4 pieces of Debris. To determine where it falls, drop four coins from knee height above the building. Place the Debris Cargo directly where the coins landed. If they landed on a unit, that unit suffers a hit.
Infantry can attack other Infantry in a building normally. But Infantry outside the building must Ruin it first somehow.
Airstrip: Planes that end movement on an airstrip can choose to take off or land. A Ruined airstrip destroys a plane that tries to use it if the plane's player flips tails on a coin.
Hangar: Planes can enter this building. By doing so they land and become ground units. Any bombs dropped while on land explode directly under the plane. It contains a stack of 2 Ammo Boxes.
Hospital: This building contains a stack of 6 Medical Supplies.
Armory: This building contains a stack of 6 Ammo Boxes. It's also hardened so it flips two coins and is only Ruined if both come up tails.
Tank Depot: Tanks can enter this building. It contains a stack of 2 Ammo Boxes.
Subway Station: Infantry ending their turn in this building can choose to move immediately to the next station in the circuit. Ruined Subway stations still work, but do not provide cover from outside attacks.
Anti-Aircraft Gun: If an Infantry unit is adjancent, and an enemy plane comes within 6 inches, the gun will automatically fire. If more than one team has infantry touching the gun, the gun does not fire upon anything. If the gun fires, all planes within 6 inches are in danger. The enemy planes are all hit and destroyed on a one-coin tail flip. Friendly and neutral planes clip two coins and are destroyed if both come up tails.
Bridge: Must be placed over water and have land connected to it on both ends. All ground units can move across it as if it were land. Hypothetical water units can move under it. When Ruined, it cannot be traversed by ground or water units.
To expand those, consider adding water vehicles, train stations and tracks, terrain rules, and other buildings. I didn't add fuel, because that seemed too fiddly. You could simplify it a lot by removing ammo but I felt that would be too simple.
It also needs scenarios with win conditions / point values for objectives. And either troop lists or a number of resource points you have to spend to buy different troops.
You could create custom scenarios by making up rules for how players take turns placing terrain and buildings to construct a map. I'd want to start with each player's units inside a deployment zone that's the same for every player, so they don't just take turns building an impenetrable base for themselves.