Let me first digress by expressing my own personal sympathies for Israel's difficult position. After returning full control of the Gaza strip to Palestine, Hamas used a civil war to wrest power, and has been using it as a safe haven to accumulate rockets from Iran and other sources to launch into Israel constantly, even during a six-month ceasefire in which Israel was relatively disciplined. Israel provided the majority of Gaza's fuel, food, electricity, and medical aid at no cost--despite access to the Mediterranean and Egypt, Gaza's Hamas leadership has held Gaza's economy back--gross mismanagement and excessive, corrupt government bureaucracy have made sure that Gaza's market has not spun up at all. Hamas hands out aid received by Israel, and takes credit--keeping its popularity high among its many poor and jobless. Hamas seems to take very little flak from its own people for storing weapons in Mosques and orphanages, or by using civilian human shields as they fight Israelis. The Arab world is starting to flare up in protest, and otherwise-moderate Arab governments have to condemn Israel's invasion to halt rocket fire--after relatively patient and persistent warnings to stop, offers to negotiate, and months of restraint--as a "war crime." Luckily for the Israelis, Egypt continues to stand by them (to what extent they can), and the Arab League is unable to spin up the momentum to actually meet over the crisis. The US will block any UN resolutions to officially condemn Israel in any way. Even US Senate Majority Leader, Harry Reid, has said that a ceasefire is impossible until Hamas offers to halt rocket fire. So even if this war lasts for a few weeks, the US will not be switching teams on a dime.
So Israel is now elbow-deep in this problem--what have they actually done? According to relatively scattered reports, Israel has already cut to the Mediterranean and has surrounded Gaza City and other urban centers, as far south as Netzarim, but the Israeli Army has been disciplined enough to avoid entering any major urban areas, where the Hamas militants likely have a great advantage (with huge numbers of civilians that the Israelis have to worry about, potential booby traps, and other "home turf" advantages like knowledge of the terrain, friendly homes, etc). A map I put together below should make it a bit more clear where the Israelis have gone.
Approximate Israeli Movement and Positions (Click for Larger)
Their staging ground was Sderot, on the northeast corner of Gaza. From there, Israel broke in during the night and cut power to most of the northern half of the strip, taking advantage of night fighting technology to strike their initial blow (and most of their advances in the last day). The Israelis are likely to continue using night to press their primary assaults in search of rockets, launch pads, and Hamas leadership. Note that with Gaza City surrounded (and certainly extremely strict policies for anyone escaping), Hamas leadership isn't going anywhere soon (for the Israelis control Gaza waters, as well). Special forces have an opportunity to use what intelligence that Mossad has gathered to take them down in the capital city if they weren't smart enough to get out early.
At this point, Israel looks like it is going to focus on a search-and-destroy mission in the north, and then perhaps move south to mop up militants that got out--it is unlikely to spread its 3 brigades so thin as to take the entire strip at once. Israel also has troops ready in Jerusalem and near Lebanon in case West Bank Palestinians or Lebanese Hezbollah militants try anything tricky to help out their brothers-in-arms.
So the easy part is over, with few casualties for the Israelis, but the hard part--search-and-destroy--is just beginning. The success of the rest of this mission will depend on Israeli tactical intelligence abilities more than brilliant ground war operational capabilities. Israeli unit tactical training will face a test, as well.
But this mission really has more substance than simply finding rockets, or even toppling Hamas leadership. After failing to meet its objectives in Summer 2006 in Lebanon--namely, preventing rockets from hitting Israel in the north--Israel is hoping to regain its deterrent capabilities with neighboring militant groups. Israel is unlikely to face a regular war with its neighbors in the near future--their willingness to negotiate has a lot to do with Israel's ability to bloody them up if they choose to fight. So far, Hezbollah has chosen to sit this one out (so far), and read (though I've lost the link) that there are rumors floating about Hamas heading to Egypt to talk potential ceasefire options. So maybe it's working.