Use the explorers ward when you're unsure of where the enemy is at the time. The most common use for it is warding a bush when enemies run into it before minions spawn. Try to get the philosopher's stone ASAP, but never go back into the lane with no wards. Even if you have enough for philosopher's stone but not enough for wards, but some wards. After philo stone, get a regular sight stone; the ruby sight stone can wait until team fights begin. Next, get boots of mobility, and don't bother upgrading them until you have your full build. These will allow you to run faster into lanes, and grab unsuspecting enemies who won't have a chance to escape. Runic bulwark is a good support item in general, as it helps blitzcrank get more tanky along with your whole team. Get shurelya's only if you will remember to use it; it does no good if the active is never used. Frozen heart is always a great item on blitzcrank, as it gives him tons of armor, debuffs the enemy adc, and it can give blitzcrank giant oversheilds. Mikael's crucible is only needed if the enemy has a lot of cc, and never get it when your adc already has cleanse. Randuins is good if the enemy has a lot of AD people. Locket of the iron solari is also good if the active is used. If any of these items are obviously not good for the situation, just replace them with tank items.

In early game, wait until you get a good chance to engage on the enemy, and then do it. Make sure not to grab at stupid times like when the enemy is level 6, and you aren't. If you are playing with a passive adc, just grab them, knock them up and retreat. this will allow your bot lane to get tons of harass, if not a kill on the enemies. If you fall behind, let the enemies push to your turret, and try to hit grabs. If you can do this, you can catch up the enemy again. Once you are ahead, feel free to zone the enemy, denying them of both exp and cs.

By this time, your adc might be roaming. You want to stick with your adc most of the time, but you can also help your team by pushing the enemy turret. Only attempt this at a perfect time, like when you are sure most of the enemies are back at the base. You can also gank the middle lane, but having your boots of mobilities is essential to doing this.

Just stick with you team, and try to catch enemies out of position. If your team splits up, try to get them to regroup, but if they don't listen, either follow your adc, spend the free time to ward, or push a lane that is safe to push. Avoid jungling because that's what allies will most likely do in the late game when the team is split.