The good news about childhood asthma is that your daughter is more likely to grow out of it. The airways of small children are narrower than in adults and they will grow as the child ages. Of course, this isn't true for everyone - your daughter may have asthma into adulthood as well.
If I were you, I'd take her to an allergist as well as the pulmonologist to investigate the possibility of allergies as triggers. If she is found to have many allergies, it would be a good idea for her to stay with the allergist since they often have more experience in treating asthma. But if she's found not to be an allergic asthmatic, you could stick with the pulmonologist.
Since she's prone to respiratory infections, the pulmonologist may get a sweat test done for her to rule out cystic fibrosis, a genetic lung and gastrointestinal (stomach) disease. But again, I'd suggest taking her to an allergist first to identify what, if any, allergens exacerbate her asthma.
You've asked about her future, but to tell the truth, no one on here can accurately predict what the future will bring. Asthma in childhood tends to be a bit more severe, so you may want to be prepared for ER visits and possible hospital admissions. As she gets older, say 10-12 years old, there are many paths her asthma could take. It could go away completely and you'll never hear another wheeze again, or it may become more severe. Another thing it could do is make you think that it went away, but may resurface later in life.
Oh, and albuterol is an asthma "rescue medication". It works by relaxing the muscles that surround the airways that get tight during an attack.
I hope that all made sense.
I wish you and your daughter the very best of luck in the future.
Daughter - 8mos. old 
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