Pros:

• Easy lane zoning
  
• All around champion for damage and tankiness
  
• Good mobility
  
• Easy to play (if you know how)
  
• Lots of damage for ADC/APC kills
  
• Snowballs very hard and easy to win lane

Cons:

• Jax can't initiate
  
• Initiating with Jax is instant death causing a lost team fight
  
• Single target damage
  
• Has to play the role right otherwise lost chance with ADC/APC kill
  
• CC is a major counter which kills Jax very hard

As said earlier, start with a Doran's Shield, Doran's Blade, or Cloth Armor. Start with Counter Strike all the time but don't max it. Once you go to lane, don't farm the first wave. Hit the enemy champ once and press Counter Strike to start the stun. After hitting the enemy champ, the minions will start to hit you causing more damage stacks on your Counter Strike. Your Counter Strike does up to 100% more damage causing it a instant burst causing either first blood or at least zoning. You have either just wasted their Health Potions, got first blood, and/or zoned them causing you to get level 2 first. This is a instant win in early game laning. If they comeback you can try to stun and Leap Strike for first blood if you didn't get it the first time. If they have an escape like Tryndamere or Wukong, max Leap Strike first for mobility to catch them for easy kills and snowballing. If you are winning lane, rush a Trinity Force or Blade of the Ruined King. For champs with escapes or tankiness, rush a Blade of the Ruined King. For anyone else, rush a Trinity Force because it is a core item for Jax and very good early game damage. If you are losing lane, rush a Randuin's Omen and play passive/safe. These options make it almost a guarantee win in lane or at least not feed or lose turret.

After getting a Blade of the Ruined King, Trinity Force, or Randuin's Omen, finish whatever you didn't get (if you got Trinity Force, finish a Blade of the Ruined King and Randuin's Omen). After that you can choose whether which boots is appropriate to buy. If you see that your bot lane is losing very bad and their ADC is very fed, get a Ninja Tabi. Otherwise just get Mercury Treads unless they have an AD Mid, AD Jungle, and AD top. If your mid is losing, you can chose to push top lane and do a quick gank mid. Ping at least twice so you know you get your mid's attention. If your mid is not low, tell them to initiate so the enemy doesn't notice you on the map. Having your teammates initiate causes the enemy to focus on killing or escaping from your teammate. This takes attention off the map causing you to come in easily without the need to worry about wards. If your mid is low, you have to come jump in so you take damage and peel for your mid. You can either take the kill and snowball yourself, or give the kill to your mid and trust them that they can make a comeback.

Get a Spirit Visage for some MR and more health. This will make you very tanky. You will last long enough in a team fight to kill the carries and possibly ace the team if played right. For the last item, get a Guinsoo's Rageblade or Guardian Angel. Get a Guinsoo's Rageblade for maximum damage and securing kills. If you are focused or you just die a lot, get a Guardian Angel. This build will make you all around tanky and have major damage. Even though you have major damage and sustain, DO NOT INITIATE WITH JAX. Jax isn't a initiator. Jax will leap in and die and get CC'd. Don't go in last either otherwise they will jump your ADC. Go in 2nd or 3rd and tank some but not all. This will make you secure kills and not die in the team fight. This will make you do a successful job in doing your job.

In lane, last hit like any other champ. Save some mana. Don't use mana for poking otherwise you will be out of mana and you will get ganked. Use mana if you know you can kill or for a giant poke (not constant little poke that makes you OOM {out of mana}). At level 6, you can hit the minions twice then hit your enemy in lane. This makes it so all your ult damage will be used on the enemy champion for major poke. You will farm more CS (creep score or minions) and poke very hard. If you do it right you can get them low enough to further zone them, waste Health Potions, and secure an easy kill. After you kill an enemy champ, don't recall unless you know their jungler is coming. If they are mid or bot push and damage turret then recall. Do not overstay or overextend. Playing safe will benefit everyone and win the game. Risking kills all the time can get you a couple kills but you will receive many deaths. This is not worth it.

Jax is an overall strong champ. He requires some skill but good for beginners too. If you play with the right mechanics like safely farming and managing mana, you will do very good and may be able to use him for elo carrying. Practice makes perfect.