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When Solo meets Greedo in the Mos Eisley cantina, he tells him:
I was just going to see your boss. Tell Jabba...
Before shooting him and walking out. He doesn't go and see Jabba, so what was he doing on Tatooine, the power-centre of the Hutt's empire? It seems like an unlikely choice for a restful drink if you're avoiding the local crime lord.
This time period is addressed in the book Rebel Dawn, which is now Legends Canon. This has not yet been addressed within the revised canon.
Han Solo is on Tatooine at the beginning of A New Hope because he just recently tried to meet with Jabba to discuss the fact that he had been boarded by Imperials and had to jettison his cargo of glitterstim. Jabba, however, was unresponsive and unwilling to meet with Han.
Han was still attempting to get a meeting with Jabba to avoid conflict, or to acquire enough money to pay off Jabba. Jabba, however, decided to send bounty hunters to collect the money or Han's corpse.
Until Greedo showed up, Han was doing exactly the opposite of trying to avoid Jabba - he was specifically trying to get a chance to actually talk to him. When Greedo shows up, Han initially implores Greedo to pass along Han's message. Instead, Greedo pulls a blaster. Unfortunately for Greedo, he never gets a shot off before Han shoots him first.
With a bounty on his head, Han realizes that Jabba isn't willing to hear him out and decides to get out while he can, and accepts a fare from some old guy, a boy, two droids and no questions.
You're not wrong. The new (and fully canon) Star Wars junior novelisation gives us some good insight into Han's thought process. In short, he's taking a moderate risk hanging around the spaceport but it's not quite as bad as you might think. The bar is the most likely place for him to get the sort of high-risk, high-reward jobs that he specialises in. On top of that, he's got a good position allowing him to see the bar entrance and patrons. He's also pretty confident (and rightly so) that if he can scrape together a few thousand credits, he'll be able to pay off Jabba and get back into his good books.
Han had taken up a corner booth back in the shadows, waiting for the right job to lure him out. And, well, okay, it had the added bonus of keeping him out of sight of the people hunting his skin. But clearly Han should have been more worried about the human girl training her sights on him than he was about Jabba the Hutt.
...
Leave it to the Wookiee to cozy right up to them, chatting them up like he hadn’t just seen someone else get his fur singed. People only came into this particular cantina if they were looking to book a ship’s services and weren’t inclined to tell the Imperials about what they were transporting. Han’s specialty.
But, really, he thought, what job was going to be worth letting a lunatic with a sword anywhere near his Falcon? They’d already had to turn down two measly jobs hauling goods off-planet. The pay was bad and the cargo was hot - literally steaming. Han might have taken a blow to his rep recently, but he had some professional pride left, thank you very much. One guy had wanted him to haul bantha dung into space to fertilize some other wasteland. No, thank you very much.
I expect that Han was going to try and smooth over the problem with Jabba, explain himself, offer to pay him off, etc. Even though he is a gangster, Jabba is a Business Hutt after all and Han did not steal from him or disrespect him in any way, he just dumped the cargo when he ran into imperial trouble. He was probably going to say that by doing so he managed to avoid going to Space Jail or whatever, so could now be indentured to Jabba for as long as it took to pay off the debt.
He tried to take off without seeing Jabba, because he thought he could do this job and then give Jabba a big stack of Space Money to buy his way back into Jabba's good books.
Solo wasn't a smuggler because he had a taste for adventure; it's how he made his living. So he had to go where the work was and where his reputation was known well enough that other criminals would give him work. If Solo ran off to a place where no one knew him, he'd have to hustle for a honest living, presumably in the shipping business. This would be a pretty tough market to break into with a freighter that looked like it was falling apart just sitting on the ground.
So Solo had no choice but to return to the wretched hive of scum and villainy in search of work to pay his debts, and he found it transporting Luke and Obi-Wan. As Han said to Chewbacca, "Seventeen thousand! Those guys must really be desperate. This could really save my neck."
As Han said himself he had the money to pay Jabba... He was there to get out of trouble.
Greedo: Going somewhere, Solo?
Han: Yes, Greedo. As a matter of fact, I was just going to see your boss. Tell Jabba that I've got his money.
Greedo: It's too late. You should have paid him when you had the chance. Jabba's put a price on your head so large every bounty hunter in the galaxy will be looking for you...Lucky I found you first.
Han: Yeah, but this time I've got the money.
Greedo: If you give it to me, I might forget I found you.
Han: I don't have it with me. Tell Jabba ---
Greedo: Jabba's through with you. He has no time for smugglers who drop their shipments at the first sign of an Imperial cruiser.
Han: Even I get boarded sometimes. Do you think I had a choice?
Greedo: Tell that to Jabba. He may only take your ship.
Han: Over my dead body.
Greedo: That's the idea. I've been looking forward to this for a long time.
Han: I'll bet you have.
Actually, Lucas had intended to include a scene in which Jabba meets Han at the Falcon before it leaves. It wasn't included in the original release, presumably because Lucas couldn't produce a convincing Jabba with the special-effects technology of the time. Much later, CGI caught up, and Lucas used it to add Jabba to some previously-discarded footage of the relevant scene, which was added in a re-release.
For the record, I don't believe Jabba put a bounty on Han until AFTER he didn't come back from ferrying Luke et al (but I admit I have no canon to back this).
