Four babies and I used convertable carseats with all of them...never had a baby bucket (infant carseat). I never saw the need for one. I'm a big believer in holding little babies as much as possible and I see way too many babies who seem to spend their whole lives in the bucket. I didn't want mine to sleep in a carseat anyway...I much preferred them to sleep in their bed and develop good sleeping habits at a very early age...Day 1 actually. Convertable carseats are more cost efficent too, because they can last for 4-6 years. If you have a large baby, they can outgrow an infant carseat as early as 4 months. You'll have to buy a convertable carseat anyway, so why not just skip the bucket and go straight to the convertable.
Even though I held them early and often, I did like having a stroller. There are times when you just can't hold them any longer and it's nice to be able to roll them. I used full-size carriage-type strollers that fully reclined so they could sleep lying flat. They still folded up to go in the car, and weren't nearly as large as those travel system thingies. Peg Perego makes nice ones. Umbrella strollers are good for compact spaces, but they don't often recline fully for young babies. The carriage-strollers are great for big toddlers too...they can nap lying down. So, one stroller can get you through the entire baby years...even through several babies. A fully-reclined stroller is a good place to change a diaper too. Very handy.
The other thing I really liked was a jogging stroller. I had a jogger with 20-inch wheels that was fantastic for outdoor activities...all-day faires, beach, rough terraine, etc. Believe me, one trip to a renaissance festival with muddy fields will convince you that you need a jogger even if you don't jog. If you get one, get the largest wheels you can find. A 20-incher will get you through the softest Jersey Shore sand...and will hold all your beach crap too. Every time I go to the beach now, I miss my jogger. I still have it, but my kids would die of embarrassment if I dragged it out now.