Colosseum tickets are very tough to acquire. You'll need to be awake a few minutes before tickets are due to go on sale for the date and time you're hoping for. There's information about the ticket-release schedule in earlier posts in this forum. If I'm not misremembering, it's a rolling release schedule, so another block of tickets goes on sale every 15 or 30 minutes. I'd start by trying for as early a ticket as would work for my schedule. If you miss that ticket, just stay online and try when the next block goes on sale.
It would be very smart to walk through the purchase process a day or more ahead of time, so you'll know what to expect.
When you're ready to purchase, have all your credit cards and debit cards right in front of you, because you never know when a European ticket vendor will reject a US card.
I'd plan to hit the Colosseum fairly early during the stay in Rome. If you are unable to get a ticket on that day, you'll have opportunities on the following days. However, you should consider how you respond to the jetlag and sleep-deprivation of an eastbound transatlantic flight; you don't want to sleepwalk through the Colosseum and have no memory of the experience. Your first full day in Rome might not be the right time for you. (It wouldn't be for me.)
The Underground ticket is perhaps the very hardest Italian sightseeing ticket to snag. Think ahead of time about how willing you are to compromise and accept a different ticket, because (from my reading of this forum) your chances of buying an Underground-access ticket are very small. You don't want to miss out on the Colosseum completely just because you can't get into the Underground.
Plenty of folks have been willing to pay a bit extra for tickets including tours (even tours in languages they do not speak), because the cheaper, plain-entry tickets sold out before they could buy them. I think you'll find tours on the Colosseum website are somewhat less expensive than tours run by commercial companies.
Be aware that private companies will probably sell you Colosseum entry before the tickets are actually on sale. They are hoping they'll be able to buy the tickets they need, but that sometimes doesn't work out. On those occasions the tour company contacts the customer and says, "So sorry." There's not much a customer can do at that point, because all the (less expensive) entry options on the Colosseum's own website are sold out.
If you're interested in the Vatican Museums and/or the Borghese Gallery, those are other tickets that need to be purchased well ahead of time. Fortunately, they don't seem to be as challenging as the Colosseum tickets. However, you're traveling to Rome during Holy Year, and April is a busy month in that city even in normal years. I'd recommend looking at the ticket situation at the Vatican Museums and the Borghese Gallery for visits earlier than you plan to be in Rome. What does availability look like at your wake-up time on the day the tickets go on sale? Can you afford just to take care of those tickets after breakfast, or do you need another early wake-up call? I didn't have to set an alarm when I bought my tickets, but I was not traveling in April or during a Holy Year.
The Vatican Museums have a lot of ticket options, so--if you're interested in seeing them--you should explore the offerings ahead of time and establish a list of preferences. I think if you buy really early you'll be able to get a plain entry ticket, but tickets including tours don't seem to sell out quite as early. (Things might be different during a Holy Year, I guess.)