(The tl;dr version: keep trying. Take a break, then try again. And if that doesn't work, try again.)
Babycenter would seem to indicate that you're doing all of this correctly:
You can introduce solids any time between 4 and 6 months if your baby is ready. Until then, breast milk or formula provides all the calories and nourishment your baby needs and can handle. His digestive system simply isn't ready for solids until he nears his half-birthday. [...] Introduce other solids gradually, one at a time, waiting at least three days after each new food. This way you'll get a heads-up if your baby has an allergic reaction to one of them
So, it seems you're doing the right thing! Now, your other questions are a little more detailed. The link I provided you above should address most of your concerns, but the relevant portions I'll quote directly.
It seems most children are initially disinterested in solids; chances are you're offering her just the right amount:
If your baby doesn't seem very interested in eating off the spoon, let him smell and taste the food or wait until he warms up to the idea of eating something solid. Don't add cereal to your baby's bottle or he may not make the connection that food is to be eaten sitting up and from a spoon.
Begin with a once-a-day feeding, whenever it's convenient for you and your baby, but not at a time when your baby seems tired or cranky. Your baby may not eat much in the beginning, but give him time to get used to the experience. Some babies need practice keeping food in their mouths and swallowing.
Once he gets used to his new diet, he'll be ready for a few tablespoons of food a day. If he's eating cereal, gradually thicken the consistency by adding less liquid. As the amount your baby eats increases, add another feeding.
(Emphasis mine, above.)
Questions about food texture have been asked before, but I'll reiterate the advice.
Keep in mind that it is normal for babies to balk the first time — or the first many times — they experience food other than breast milk, formula, or liquidy purees. That's why it's important to keep offering different foods to babies who are developmentally ready — especially healthy ones like vegetables. [...]
Textures take getting used to, just like tastes. "Kids often don't like the texture of avocado but are won over by the taste," Altmann says.
So, keep offering your child the solids. It won't be a problem until she's about a year old. If she hasn't transitioned to solids by then, then you ought to consult with your healthcare professional.
Finally, if you're willing to take a "risk," offer her some, ah, more adventurous options instead.
- Sour fruits (cherries, plums)
- Stewed meat (sometimes even spicy!)
- Cruciferous veggies (cauliflower, turnips)
- Whole grains (quinoa, millet)
For more information, check out this feeding guide. Also, the comments to this question may be enlightening, though you might find this thread more relevant to your experience.
Finally, What To Expect has a lot of information about the transition to solids, and they're kind of the go-to source for most "westerners" in my experience.