One way to see Barcelona for the first time is on the 'Bus Turistic' Bus tour (althought you'll have to do long queues to get in such a tourist trap). They depart from Plaça Catalunya (where the airport bus leaves you). It is best if you pick up a route map and take it to one of the many cafes, etc. and have a look through it to decide if there's any places of specific interest you want see.
You can buy daily tickets and if memory serves correctly tickets are valid for 2 days. These are best because there are 3 routes (2 during winter) which last for quite some time, especially if you decide to get off and visit some of the sights around the city. You can hop on and off the bus as you like, when you have finished at one location just get on the next that comes a long.
With the booklet you get onboard the bus there will be vouchers, which give you discounts to various places the tour visits. If you are visiting Barcelona on a budget the tour bus can be expensive and if you don't want to follow the crowds you might be better off buying a T 10 travel ticket and a map. Barcelona is not a big city and the public transport is good so take advantage of it and make your own way around.
One of the best ways to understand Barcelona and to soak up the atmosphere is to go for a stroll throught the streets of the Old Quarter (Barri Gòtic). La Rambla is great and you should definitely check out the street entertainers, take your time, browse in the shops etc but, don´t be afraid to turn down the side streets, you never know what gems you will find. Most of the best value, best tasting and most authentic tapas bars and cafes are in the winding streets and little squares (places).
If you're coming to Barcelona for the first time with Children there are plenty of things to do. The Barcelona Aquariam is excelent for children of all ages. Barcelona also has a big Zoo even with a dolphin show,. Tibidabo is a theme park on the top of Tibidabo hill and is amazing fun.
When in Barcelona, especially in the touristy areas ALWAYS WATCH YOUR POSSESSIONS!! Never underestimate the tenacity and sheer cheek of Barcelona's many pick pocketers and con artists. Don't stand around in the middle of the street with your bag undone, map in one hand and guide book in the other because they will spot you a mile off. Just be careful. Don't take all your ids, cards and cash with you, keep some in different locations for emergencies. Don't fight back if they do take things, but don´t expect to get it back again either. That is, unless you meet some of the cheekiest pick pocketers of all who will swipe your stuff then come back a few moments later and offer to sell your wallet back to you minus the cash contents (leading you to go to a cash point to get more......can you see where this is going?)
Pick pockets in Barcelona come in all shapes and sizes in Barcelona, including children. So be on the look out all the time especially wheh in main tourist areas and in the metro.
Don't be put off by this warning though, Barcelona is a great city and recommended for anyone.
Enjoy!
