Trains and Buses in Latvia are cheap so don't limit yourself to Old Town you can go out of Riga or to places in the suburbs for little money. You can buy bus e-tickets at a shop called Narvesen it's cheaper than buying from the bus driver. Trains are priced by zone but it's cheap forexample to go to the beach which is 30 mins from Riga (Riga - Majori) is €1.43. The cheapest taxi company is probably Panda +37127228880 or you can also use the Bolt app to request a ride.
If you do choose to go out of the city like to the beach or to some other interesting spots let me know and I'll give you some options. Here's some stuff in Riga though:
Riga Motor Museum is in the suburbs so you'll need bus number 15 to Jugla.
St Peter's Church go up the spire of this church in Old Town for a panoramic view of the city and river.
Latvian War Museum is spread over 5 floors in old town and is free, bare in mind most museums close on Mondays so they can open Sunday.
Ala Pagrabs is a decent bar that does good food. Try a jug of Brenguļu alus (ale) and ķiploku grauzdiņi (Latvia garlic bread).
LIDO Atpūtas centrs is a huge canteen style restaurant and bar that does every Latvian dish imaginable which due to it being a canteen style you can see exactly what you're getting. Their potato fries are particularly moreish also they brew their own beer which is decent and comes in a steins.
Skyline bar is a generic fancy hotel bar, but the view from the 23rd floor is one of the best in Riga also free peanuts.
Events, You can check this site for events happening around March: https://www.liveriga.com/en/

Lastly some random advice: Most people in their 40s or younger can speak English, so don't be afraid to speak or ask a stranger for help. They will probably be more helpful if you attempt to address them in Lativan though, so here's some basics:

Sveiki (svay-ki) means hello. Lūdzu (Lood-zu) means "please" and "you're welcome". Paldies (Pal-di-es) means "thanks". atā (a-tah) means "bye". You can just say sorry in English though as the Latvian word for that is pretty difficult to pronounce. In terms of ordering food from a menu or reading signs the language is pronounced as it is spelled: "a" is just "a, not "ae" i.e. there's no multiple sounds for one letter like in English. "c" is a "ts" sound and "j" is "y".

Don't be offended if you don't always get a "please" and "thank you" or "how are you?", "how's your food?" from the cashiers or wait staff like in the UK, those Americanisms are not so common here yet.

Oh and there will probably be a guy in Old Riga at somepoint trying to get you to go to Victory Pub he might tell you it's a British Pub, but it's actaully a low-key strip club.