Some good suggestions so far. I will throw in my two cents as well as throw a few more names in the mix:
Elizabeth: Great option; casual fine dining. Comfortable environment for someone new to tasting menus; very relaxed, friendly atmosphere with no stuffiness and some outstanding food as well as beautiful plating, whimsy, and unique dishes/ingredients. The chef/owner is a wonderful person and has a great staff.
Alinea: A bit over your stated price range and IMHO not the best option. While up until recently Alinea has been Chicago's clear cut #1 restaurant, I feel as though recently they have been slipping. Chef Achatz has been spending considerably less time in house and he and his partner have been focused on expanding their empire and boosting profits rather than putting the customer first and trying to continue to keep improving Alinea. I am hoping this is a temporary step backwards and not a sign of Alinea being past her prime. I have dined here five times and my first four were among my best meals ever, but my last was disappointing; fast pace (was in and out in just over two hours versus approximately four hours during previous visits), subpar service with some amateur mistakes and lack of attentiveness, and a lack of innovation compared to prior meals. It also can be a bit intimidating of an environment for a first tasting menu experience and is not nearly as relaxed or friendly as Elizabeth. Still an amazing restaurant, but no longer the clear cut best and with the reduction in courses and service and increased cost it is arguably a poor value (if you add wine pairings the cheaper pairing is $150 per/person before tax/tip - so dinner for one with the lesser pairing comes to approximately $500 - though you can save some money by doing a bottle or some wines by the glass).
El Ideas: Very fun, lively atmosphere - kind of like being at a dinner party. You can hang out in the kitchen between courses. The food is as amazing and creative as any fine dining venue, but the atmosphere is totally relaxed. There is only one server; the chefs bring out and present the courses. Might not be the type of experience you are seeking (if you want a more traditional fine dining meal), but worth looking into. Keep in mind they do very limited substitutions, so not a good option if either of you have significant dietary restrictions.
Grace: In my opinion Grace will be challenging Alinea for the #1 spot in the next year or so; personally I think they may have already eclipsed Alinea but it will take a little time for this to become widely regarded. Nice blend of traditional and modern fine dining but stellar, personalized service that makes the experience relaxing. Some of the best tasting food you will ever taste and beautiful, creative plating. Stunning. comfortable dining room. With a wine paring it will come to about $380, so just a tad over your price point.
Moto: Somewhat similar style of food as Alinea (known for molecular gastronomy) but the current executive chef (Richie Farina) has moved them a bit away from this and has focused on making the food outstanding while still retaining some of the whimsical, scientific experience Moto was known for (of late they generally only use molecular gastronomy if it enhances the food rather than just for show). The pastry chef (Claire Crenshaw) who came over from Tru really elevated their desserts. Great staff. More affordable than Alinea (about $300 with pairings) and while Alinea I feel is slipping, Moto is ascending.
Goosefoot: This is worthy of being a top choice, but has become the hardest reservation in Chicago - so you will likely have to work hard for it if you want to diner here (they are on Open Table, but reservations get snagged as quick as they are released, so you have to get lucky and be looking shortly after someone cancels). The food is fine dining at is best (my choice for best tasting food in Chicago) but the atmosphere is very laid back and far more casual than venues such a Grace or Alinea. Service is friendly, but not as polished as at Grace and Alinea. My only complaint about Goosefoot is how seldom the menu changes significantly, but since this would be your first time it would be a non issue. Nice relaxing environment for a first time tasting menu. Like El Ideas they are BYOB, so at $135 plus tax/tip well within your price range.
Senza: Similar to Goosefoot in terms of fine dining but a relaxed environment and very friendly staff, but not as polished as Grace, Alinea. They do have wine program (and some great cocktails too). The executive chef came over from Schwa and one can see the influence of Schwa in his dishes (not a good choice if you like a sweet component to your savories; like Schwa there is a lot of mixing of sweet and savory - thought not to the same extreme). Also a very comfortable locale for a first time tasting menu experience; unlike Goosefoot reservations are easy (they also are on Open Table). Not as well known as other fine dining venues (despite their recent Michelin star) but all my meals here have been terrific. While they are gluten free; if you were not told this you would never know (their bread and pasta here are among the best I have had anywhere).
Sixteen, Tru and L2O are also all great venues, but their longest menus with pairings would be a bit over your stated price point and also I do not think they are the best atmosphere for a first time fine dining experience (and at Sixteen I have experienced some consistency issues in both food and service). Schwa would be within your price point, but their tendency to cancel reservations and short notice as well as their crazy, wild ambiance would likely not be a good choice for a first tasting menu experience; when Schwa is on it is among the best food anywhere - but definitely a wildcard in terms of what you will experience on any given night.
Twitter: @Goof_2