Both A2 and Grand Rapids are about an hour away from Lansing. Detroit a bit more. If you don't want to go anywhere, the best spot in Lansing to eat is probably Red Haven, which is on the east side of East Lansing, just past the edge of the MSU campus. I haven't been anywhere else in years though, so there could be other interesting things happening.
But if you're game for that commute and you head to Ann Arbor...
Spencer: Cute spot, communal seating, short menu, very "natural" focused wine list heavy on Massale and Dressner IIRC. This is the only one of my suggestions that I've been to in the last 3 years.
Aventura: Spanish tapas, mostly. Food was, in years past, solid. The big advantage here is that they have always had a large selection of wines from one of our Michigan-based importers, Vino Vi & Co, which has some really great small Spanish producers in their book. I've seen a few of their wines show up in Chambers St mailings of late, but many of the wines are only available through her in the US, so a good chance to maybe try a few things not widely available outside Michigan.
Last Word: Cocktail spot. If you go on certain weekdays, they run particular deals. Used to have a half off night for whiskey pours on Wednesdays, which makes some pretty outstanding, expensive things a more affordable proposition.
Kosmo's: Small Korean bibimbap spot in a little cluster of shops near Zingermans Deli.
Everyday Wine: Across the hall from Kosmo's. Value focused, but they usually have a few interesting things, and Jackie and Mary are just two really nice, wonderful people.
Zingermans: If you haven't been, you must go. Especially if you want to browse olive oils, vinegars, cheese, etc to kill some time. Legendary for a reason, though you'll pay for the pleasure.
Tomukun: Noodle bar w/ some Korean. Closer to campus and usually quite busy at lunch.
Comet: Good coffee spot in a little covered outdoor mall area called Nickels Arcade.
Lab Cafe: Coffee, tea, and better-than-Comet's pastry selection. Next to Tomukun.
There's plenty else, of course -- especially if the college town vibe is appealing to you. Most of what I listed above is mostly a bit off campus but walkable if the weather cooperates for a stroll around town.
Not walkable would be the other wine shop recommendation, which is Morgon & York, probably about a 5 minute drive from downtown.
If you wanted a single recommendation, I'd probably say Spencer on the merits of the simple, delicious menu. But Aventura is a very quick walk, so it'd be easy to hit both if you had some time.
If you are at all entertaining the idea of coming to Detroit, there are quite a few options. Not sure how often you'll make it out during your stay, but if you think you might come this far, let me know and there's a much bigger, more interesting list than there would have been even 2 years ago.