If you plan to breast feed then its recommened to solely BF for 6-8 weeks to a) establish your supply and b) to prevent nipple confusion. They say to express and bottle feed it only once this time is past and breastfeeding is well established. Mixing the two can cause problems to you and baby so I'd say its better to wait.
I wanted my OH to be able to feed our LO but didn't express till my supply was well and truly established and LO was feeding really well. It meant he was not able to in the early weeks but that was ok. As midna said, baby will spend so much time on your boob and its important they do so to establish a good flow and supply.
Expressing often gets very mixed results and some women can't express much at all but direct from the boob baby gets plenty. Others can express enough for their LO's.
As for nipple confusion... babies suck differenly on the nipple to a bottle. If your LO finds it hard to latch in the early days you'll need to keep persevering till they get the hang of it. Giving milk from a bottle too soon can mean they get lazy as often from a teat it flows much quicker and they don't have to suck to get the milk. So when they go back on the boob they don't work so hard to get the milk out and this can affect your supply build up.
I'd strongly suggest that you stick to your boob for the first 6 weeks and get the supply going based on your LO's demands. Then once established express the odd feed or two for your OH to give. You'd still have to express when your OH feeds anyways as otherwise it will mess up the supply and demand from your boob in the early weeks. Its not like you can take time off from a feed as your boobs alter demand based on what is taken out of them. So if you express a feed and your OH does it, if you then don't express from your boob, your body will adjust accordingly and maybe not produce so much milk. As hard as the first few weeks are, if breastfeeding I'd say stick to the boob as far as possible, if not solely.
I'd personally not mix formula with breastfeeding unless you have real supply problems. Giving formula so your OH can do some feeds would again a) possibly cause nipple confusion b) upset LO's tummy if not able to digest the milk properly or it does not agree c) mess with your milk supply and cause problems there for you and LO.
I did try expressing at 3 weeks on advice from my HV as I had a *very* demanding baby and health problems since the birth and was a wreck. My LO had never had any problems latching etc and my supply was in abundance. However, I expressed about 2 feeds and thought to hell with this and gave up. I had to express when OH fed LO regardless, it didn't mean I got any extra sleep as LO wanted Mummy to cuddle him back to sleep and would not settle etc in basket... the list goes on. It was easier to feed him off my boob, cuddle and fall to sleep together.
Once the 6 weeks were up OH tried bottle feeding expressed milk but didn't actually find it very enthralling or bonding and said he got more from cuddling and holding our son and interacting with him than he did feeding him. He and I were more than happy we stick to boob feeding. I do express off some milk still and freeze in case its needed but otherwise I boob feed only.
As for being shy, invest in a shawl if you really can't feed in public. TBH I doubt you'll mind once LO is here. I certainly don't and am happy to feed out and about if need be. Most people take no notice.
i know it sounds stupid, and im sure when the time comes i wont be worried about it as much. but the thought of it makes me not want to breast feed. im just way too much of a private person 
anyway, if people came to visit in the early days, i'd go off to our bedroom to feed, and if out and about i took bottles of expressed milk to feed him - connor's first feed was formula from a bottle as he could feed from my boobs to start with. but as others have said, try and avoid this in the first 6-8 weeks cos of nipple confusion - it might not be a problem for you (like it wasn't for us) but its not something you want to have issues with if you can help it. 
